44.07.03 · optimization-control / optimal-control-pontryagin

Bang-Bang and Minimum-Time Optimal Control

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Anchor (Master): Pontryagin, Boltyanskii, Gamkrelidze & Mishchenko 1962 The Mathematical Theory of Optimal Processes (Interscience/Wiley) Ch. 2-3 (the maximum principle for time-optimal linear systems, the bang-bang theorem, the number of switches bounded by the eigenvalue structure, the harmonic-oscillator example); Hermes & LaSalle 1969 Functional Analysis and Time Optimal Control (Academic Press) Ch. 2-4 (existence via Filippov, the bang-bang principle as an extreme-point/Lyapunov-convexity statement, normality and uniqueness); Athans & Falb 1966 Optimal Control (McGraw-Hill) Ch. 7 (synthesis of the time-optimal feedback and the switching curve)

Intuition Beginner

Picture a toy car on a straight track, and your job is to get it to a marked spot and stop there in the least possible time. The only controls you have are the gas and the brake, and both are all-or-nothing: you can push the car forward at full force, or push it backward at full force, but there is no gentle in-between. The question is when to switch from one to the other.

The answer is wonderfully blunt. To reach the spot fastest, you slam the car forward at full force for a while, then at exactly the right moment you slam it backward at full force to kill the speed, arriving at the spot just as you come to rest. One hard push, one hard brake, and a single switch between them. Any plan that eases off the pedal partway wastes time, because half-force never beats full-force when the clock is running.

This all-or-nothing, switch-once behaviour has a name: bang-bang control. The "bang" is the control pinned to its hardest setting; the two bangs are the two extremes, full forward and full backward.

The reason it happens is the same one behind every constrained optimal-control problem. When the cost grows steadily with the control and the control is capped, the best setting is always pinned to an edge of what is allowed, never in the soft middle. The only real decision left is the timing of the flip.

Visual Beginner

The picture is the time-optimal trip for the toy car, shown as the two phases and the single instant where the control flips.

Read the table top to bottom as the story of one fastest trip. The control is at one extreme, then flips once to the other extreme. The flip happens at the switching time, the single moment the whole plan turns on. Before it, you are speeding up toward the target; after it, you are braking so as to arrive exactly at rest.

phase	control setting	what the car does
phase 1	full force toward the target	builds up speed
switch	the control flips, all at once	the single decision
phase 2	full force the opposite way	brakes to a stop at the target

The takeaway: a hard-looking question — out of all possible driving plans, which is fastest? — collapses to a single number, the time at which you flip from full-forward to full-backward. Bang-bang control turns planning a whole trip into choosing one switching instant.

Worked example Beginner

Take the cart-on-a-track problem with numbers. The cart has a position and a speed. Pushing the cart changes its speed: the push $u$ is capped between $- 1$ and $+ 1$ , and the speed changes at the rate $u$ . The position changes at the rate of the current speed. The cart starts at position $- 2$ , at rest (speed $0$ ), and we want to bring it to the origin, at rest, in the least time.

Step 1. Decide the first push. The cart is to the left of the target and not moving, so we push right: $u = + 1$ . The speed climbs from $0$ , and the cart moves right, gaining ground toward the origin.

Step 2. Recognise we must brake. If we kept pushing right, the cart would overshoot the origin with speed to spare. So at some moment we flip to $u = - 1$ to bleed off speed and arrive at rest.

Step 3. Use symmetry to find the flip. Pushing right then braking left with equal force is symmetric: the speeding-up phase and the slowing-down phase take equal time. Call each phase length $t_{1}$ . During phase 1 the speed reaches $t_{1}$ , and the distance covered is $\frac{1}{2} t_{1}^{2}$ . During phase 2 the cart covers another $\frac{1}{2} t_{1}^{2}$ while the speed falls back to $0$ .

Step 4. Match the distance. The total distance is $2$ (from $- 2$ to $0$ ): $\frac{1}{2} t_{1}^{2} + \frac{1}{2} t_{1}^{2} = t_{1}^{2} = 2$ , so $t_{1} = 2$ . The flip happens at time $2$ .

Step 5. Total time. Two phases of length $2$ give a total time $22 \approx 2.83$ .

What this tells us: the fastest plan is $u = + 1$ until time $2$ , then $u = - 1$ until time $22$ , arriving at the origin at rest. One switch, two extremes. The whole solution is the single switching time $2$ .

Check your understanding Beginner

Exercise (easy, multiple choice).

In a minimum-time problem where the control is capped between $- 1$ and $+ 1$ and the cost is the elapsed time, what does the optimal control typically look like?

A. A smoothly varying value that eases between the limits. B. A value pinned to one of the two extremes, flipping between them at certain instants. C. The middle value $0$ for the whole trip. D. A value chosen at random each instant.

Hint

When the clock is running and the push is capped, half-force never beats full-force; the only question is the timing of the flip.

Answer

B. A value pinned to one of the two extremes, flipping between them. This is bang-bang control: the time-optimal push sits at $+ 1$ or $- 1$ and switches between them, never resting in the middle. Feedback-correct: full force shortens the trip, so the optimum lives on the boundary of the allowed range. Feedback-wrong: a smooth interior value (A, C) wastes time, and the control is far from random (D) — it is a clean two-level signal with a few switches.

Formal definition Intermediate+

Consider the linear control system $\overset{x}{˙} = A x + B u$ of 02.06.03, with state $x (t) \in R^{n}$ , control $u (t) \in R^{m}$ , $A \in R^{n \times n}$ , $B \in R^{n \times m}$ , and the control constrained to the cube

U = [- 1, 1]^{m} = {u \in R^{m} : ∣ u_{i} ∣ \leq 1, i = 1, \dots, m} .

Admissible controls are measurable functions $u : [0, T] \to U$ . The columns of $B$ are written $b_{1}, \dots, b_{m}$ .

Definition (minimum-time problem). Given an initial state $x_{0}$ and a target $x_{T}$ (often the origin), minimise the terminal time

J (u) = T = \int_{0}^{T} 1 d t

over all admissible controls steering $x (0) = x_{0}$ to $x (T) = x_{T}$ along $\overset{x}{˙} = A x + B u$ . The running cost is $L \equiv 1$ and the terminal time $T$ is free.

Definition (control Hamiltonian and costate). Following 44.07.01 with $L = 1$ , $f = A x + B u$ , the Hamiltonian (minimisation convention) is

H (x, u, p) = 1 + p^{T} (A x + B u) = 1 + p^{T} A x + p^{T} B u,

with costate $p (t) \in R^{n}$ obeying the adjoint equation $\overset{p}{˙} = - \partial H / \partial x = - A^{T} p$ . The adjoint is the linear ODE $\overset{p}{˙} = - A^{T} p$ , so $p (t) = e^{- A^{T} t} p (0)$ in the matrix-exponential form of 02.06.03.

Definition (switching function). The switching function is the vector

σ (t) = B^{T} p (t) \in R^{m}, σ_{i} (t) = b_{i}^{T} p (t),

the costate-projected coefficient of the control in $H$ . The Hamiltonian's $u$ -dependence is the single term $σ (t)^{T} u$ .

Definition (bang-bang control). A control is bang-bang on $[0, T]$ if for almost every $t$ each component $u_{i} (t)$ equals $+ 1$ or $- 1$ , i.e. $u (t)$ takes values at the vertices of the cube $U$ . The pointwise minimisation of $H$ over $U$ — minimising $σ (t)^{T} u$ over $u \in [- 1, 1]^{m}$ — separates by component and gives

u_{i}^{⋆} (t) = - sign (σ_{i} (t)) whenever σ_{i} (t) \neq = 0.

A switch of component $i$ is a sign change of $σ_{i}$ ; the control flips between $+ 1$ and $- 1$ there.

Definition (singular arc; normality). Component $i$ has a singular arc on a subinterval where $σ_{i} (t) \equiv 0$ , since then the minimisation does not determine $u_{i}$ . The single-input pair $(A, b)$ is normal when $σ (t) = b^{T} e^{- A^{T} t} p (0)$ has only isolated zeros for every $p (0) \neq = 0$ ; equivalently the controllability matrix $[b A b \dots A^{n - 1} b]$ has full rank $n$ (the Kalman controllability condition of 02.06.03's reachability circle). Under normality there are no singular arcs and the bang-bang control is determined almost everywhere.

Notation: $A, B$ are the system matrices, $b_{i}$ the columns of $B$ ; $U = [- 1, 1]^{m}$ the control cube; $p$ the costate; $σ = B^{T} p$ the switching function; $u_{i}^{⋆} = - sign (σ_{i})$ the bang-bang control; $e^{- A^{T} t}$ the adjoint state-transition matrix; $sign$ the scalar sign function.

Counterexamples to common slips

Bang-bang is not the same as "few switches." The bang-bang property — control at the cube vertices almost everywhere — holds for every normal time-optimal problem, but the number of switches is bounded only under extra spectral structure. If $A$ has complex eigenvalues (an oscillator), the switching function is a damped or undamped sinusoid and the control can switch arbitrarily many times as the horizon grows; the at-most- $(n - 1)$ bound needs $A$ to have only real eigenvalues.
Normality is a genuine hypothesis, not a formality. Without controllability of $(A, b)$ the switching function can vanish on an interval, producing a singular arc where the maximum principle leaves $u$ undetermined; the time-optimal control may then fail to be bang-bang or fail to be unique. Asserting bang-bang for an uncontrollable single-input system is the typical error.
The switching function lives on the costate, not the state. Writing the switch condition in terms of the state $x$ confuses the synthesis (the switching curve in state space) with the necessary condition (the sign of $σ = B^{T} p$ ). The switching curve is derived afterward by eliminating the costate, and only for low-dimensional, explicitly solvable cases.

Key theorem with proof Intermediate+

Theorem (bang-bang principle and the time-optimal double integrator; Pontryagin et al. Ch. 2-3 ^{[source pending]}; Athans-Falb Ch. 6-7 ^{[source pending]}). For the minimum-time problem $\overset{x}{˙} = A x + B u$ , $u (t) \in [- 1, 1]^{m}$ , with free terminal time and the maximum principle of 44.07.01, the optimal control satisfies $u_{i}^{⋆} (t) = - sign (σ_{i} (t))$ with $σ (t) = B^{T} p (t)$ and $\overset{p}{˙} = - A^{T} p$ ; in particular $u^{⋆}$ is bang-bang wherever each $σ_{i}$ is nonzero. For the double integrator $\overset{x}{˙}_{1} = x_{2}$ , $\overset{x}{˙}_{2} = u$ , $u \in [- 1, 1]$ , steering to the origin, the switching function is affine in $t$ , so the time-optimal control switches at most once, taking the values $\pm 1$ , and the loci of single-switch trajectories assemble into the parabolic switching curve $x_{1} + \frac{1}{2} x_{2} ∣ x_{2} ∣ = 0$ .

Proof. By the maximum principle, the optimal control minimises $H (x^{⋆}, u, p) = 1 + p^{T} A x^{⋆} + p^{T} B u$ over $u \in U$ for almost every $t$ . The only $u$ -dependent term is $p^{T} B u = σ^{T} u = \sum_{i} σ_{i} u_{i}$ , a linear functional of $u$ that separates across components. Minimising $σ_{i} u_{i}$ over $u_{i} \in [- 1, 1]$ sends $u_{i}$ to $- 1$ when $σ_{i} > 0$ and to $+ 1$ when $σ_{i} < 0$ , i.e. $u_{i}^{⋆} = - sign (σ_{i})$ wherever $σ_{i} \neq = 0$ . The adjoint equation $\overset{p}{˙} = - A^{T} p$ has solution $p (t) = e^{- A^{T} t} p (0)$ , so $σ (t) = B^{T} e^{- A^{T} t} p (0)$ is an analytic (indeed exponential-polynomial) function of $t$ ; under normality its zeros are isolated and the bang-bang control is determined almost everywhere. This proves the first claim.

For the double integrator, $A = \begin{psmallmatrix} 0 & 1 \\ 0 & 0\end{psmallmatrix}$ , $b = \begin{psmallmatrix} 0 \\ 1\end{psmallmatrix}$ . The adjoint equations are $\overset{p}{˙}_{1} = 0$ and $\overset{p}{˙}_{2} = - p_{1}$ , so $p_{1} \equiv c_{1}$ constant and $p_{2} (t) = c_{2} - c_{1} t$ , affine in $t$ . The switching function is $σ (t) = b^{T} p (t) = p_{2} (t) = c_{2} - c_{1} t$ , affine in $t$ , hence with at most one zero. Therefore $u^{⋆} = - sign (p_{2})$ switches at most once, taking values $\pm 1$ . The pair $(A, b)$ is controllable ( $[\,b\ Ab\,] = \begin{psmallmatrix} 0 & 1 \\ 1 & 0\end{psmallmatrix}$ has rank $2$ ), so the problem is normal and no singular arc occurs.

To assemble the switching curve, integrate a single constant-control arc. With $u \equiv \pm 1$ , $\overset{x}{˙}_{2} = u$ gives $x_{2} = x_{2} (0) + u t$ , and $\overset{x}{˙}_{1} = x_{2}$ gives $x_{1} = x_{1} (0) + x_{2} (0) t + \frac{1}{2} u t^{2}$ . Eliminating $t$ via $t = (x_{2} - x_{2} (0)) / u$ and using $u^{2} = 1$ yields the family of parabolas $x_{1} = \frac{1}{2 u} x_{2}^{2} + const$ , opening in opposite directions for $u = + 1$ and $u = - 1$ . The two arcs that pass through the origin are the half-parabolas $x_{1} = - \frac{1}{2} x_{2} ∣ x_{2} ∣$ : the arc with $u = - 1$ arriving from $x_{2} > 0$ and the arc with $u = + 1$ arriving from $x_{2} < 0$ . Their union is the switching curve

γ = {(x_{1}, x_{2}) : x_{1} + \frac{1}{2} x_{2} ∣ x_{2} ∣ = 0},

the locus from which a single final constant-control arc reaches the origin at rest. $□$

Bridge. This sign-of-the-switching-function argument builds toward the Advanced results, where the same costate-projection $σ = B^{T} p$ governs the switch-count bound under real-eigenvalue structure and the harmonic-oscillator example whose sinusoidal $σ$ produces periodically spaced switches, and it appears again in the synthesis section, where the switching curve $x_{1} + \frac{1}{2} x_{2} ∣ x_{2} ∣ = 0$ becomes a state-feedback law. The foundational reason the minimum-time control is bang-bang is exactly the linearity of $H$ in $u$ over a polytope: the minimiser of a linear functional on a polytope is attained at a vertex, which is the central insight that this unit shares with linear programming and which generalises the scalar $u^{⋆} = - sign (p)$ of 44.07.01 to the cube. The switching function is dual to the reachable-set support functional, and putting these together, the bridge is the recognition that minimum-time control replaces the smooth interior optimisation of the linear-quadratic regulator of 44.07.02 — where $H$ is quadratic in $u$ and the optimum is interior, giving smooth feedback — with a boundary optimisation whose optimum is a cube vertex, giving discontinuous bang-bang feedback.

Exercises Intermediate+

Exercise 2 (easy, multiple choice).

The bang-bang property (optimal control at the cube vertices) follows because the Hamiltonian's dependence on $u$ is:

A. quadratic in $u$ , so the optimum is interior. B. linear in $u$ , so the minimum over the cube is at a vertex. C. independent of $u$ , so any control is optimal. D. nonconvex in $u$ , so there is no minimum.

Hint

The $u$ -dependent part of $H$ is $σ^{T} u$ ; minimise a linear functional over a box.

Answer

B. linear in $u$ , so the minimum over the cube is at a vertex. The $u$ -dependence of $H = 1 + p^{T} A x + σ^{T} u$ is the linear term $σ^{T} u$ ; a linear functional on the cube $[- 1, 1]^{m}$ attains its minimum at a vertex, giving $u_{i}^{⋆} = - sign (σ_{i})$ . The quadratic case (A) is the linear-quadratic regulator of 44.07.02, which has interior smooth feedback, not bang-bang. Rubric: full credit for B.

Exercise 3 (medium, symbolic).

For the double integrator $\overset{x}{˙}_{1} = x_{2}$ , $\overset{x}{˙}_{2} = u$ , $u \in [- 1, 1]$ , show that the switching function $σ (t) = p_{2} (t)$ is affine in $t$ , and conclude there is at most one switch.

Hint

$H = 1 + p_{1} x_{2} + p_{2} u$ ; compute $\overset{p}{˙}_{1}$ and $\overset{p}{˙}_{2}$ from $\overset{p}{˙} = - \partial H / \partial x$ .

Answer

$H = 1 + p_{1} x_{2} + p_{2} u$ . The adjoint equations are $\overset{p}{˙}_{1} = - \partial H / \partial x_{1} = 0$ and $\overset{p}{˙}_{2} = - \partial H / \partial x_{2} = - p_{1}$ . Hence $p_{1} \equiv c_{1}$ is constant and $p_{2} (t) = c_{2} - c_{1} t$ is affine in $t$ . The switching function is $σ = b^{T} p = p_{2} = c_{2} - c_{1} t$ . An affine function has at most one zero (unless it is identically zero, excluded by normality/controllability), so $u^{⋆} = - sign (p_{2})$ switches at most once. Rubric: full credit for both adjoint equations, the affine $p_{2}$ , and the single-switch conclusion.

Exercise 4 (medium, symbolic).

Derive the parabolic switching curve $x_{1} + \frac{1}{2} x_{2} ∣ x_{2} ∣ = 0$ for the double integrator by integrating a single constant-control arc $u = \pm 1$ that reaches the origin at rest.

Hint

Integrate $\overset{x}{˙}_{2} = u$ and $\overset{x}{˙}_{1} = x_{2}$ backward from the origin; eliminate time using $u^{2} = 1$ .

Answer

Run a constant-control arc backward from $(x_{1}, x_{2}) = (0, 0)$ at the final time. With $u = \pm 1$ , integrating backward by time $s \geq 0$ gives $x_{2} (- s) = - u s$ and $x_{1} (- s) = \frac{1}{2} u s^{2}$ . Eliminate $s$ : from $x_{2} = - u s$ , $s = - x_{2} / u = - u x_{2}$ (using $1/ u = u$ ), and $s \geq 0$ forces $sign (x_{2}) = - u$ . Then $x_{1} = \frac{1}{2} u s^{2} = \frac{1}{2} u x_{2}^{2} = - \frac{1}{2} sign (x_{2}) x_{2}^{2} = - \frac{1}{2} x_{2} ∣ x_{2} ∣$ . So the final arcs lie on $x_{1} = - \frac{1}{2} x_{2} ∣ x_{2} ∣$ , i.e. $x_{1} + \frac{1}{2} x_{2} ∣ x_{2} ∣ = 0$ . The branch with $x_{2} > 0$ uses $u = - 1$ ; the branch with $x_{2} < 0$ uses $u = + 1$ . Rubric: full credit for the backward integration, the time elimination, and the signed-square form.

Exercise 6 (medium, symbolic).

For the harmonic oscillator $\overset{x}{˙}_{1} = x_{2}$ , $\overset{x}{˙}_{2} = - x_{1} + u$ , $u \in [- 1, 1]$ , write the adjoint system and show the switching function is sinusoidal, so the switches are spaced $π$ apart in time.

Hint

$H = 1 + p_{1} x_{2} + p_{2} (- x_{1} + u)$ ; the adjoint $\overset{p}{˙} = - A^{T} p$ has the same oscillator structure, and $σ = p_{2}$ .

Answer

Here $A = \begin{psmallmatrix} 0 & 1 \\ -1 & 0\end{psmallmatrix}$ , $b = \begin{psmallmatrix} 0 \\ 1\end{psmallmatrix}$ . The adjoint $\overset{p}{˙} = - A^{T} p$ reads $\overset{p}{˙}_{1} = p_{2}$ , $\overset{p}{˙}_{2} = - p_{1}$ , the same rotation, so $p_{2} (t) = ρ cos (t - ϕ)$ for constants $ρ, ϕ$ . The switching function is $σ = b^{T} p = p_{2} (t) = ρ cos (t - ϕ)$ , a sinusoid of period $2 π$ . Its zeros are spaced $π$ apart, so the bang-bang control $u^{⋆} = - sign (p_{2})$ switches every $π$ time units, and the number of switches grows without bound as the required time grows. Rubric: full credit for the adjoint rotation, the sinusoidal $σ$ , and the $π$ -spacing of switches.

Exercise 7 (hard, symbolic).

Prove the switch-count bound: if $A \in R^{n \times n}$ has only real eigenvalues and $(A, b)$ is a controllable single-input pair, then each component of the time-optimal control switches at most $n - 1$ times.

Hint

$σ (t) = b^{T} e^{- A^{T} t} p (0)$ is a linear combination of terms $t^{k} e^{- λ_{j} t}$ ; bound the number of real zeros of such an exponential polynomial.

Answer

The switching function is $σ (t) = b^{T} e^{- A^{T} t} p (0)$ . Writing $A$ (hence $A^{T}$ ) in real Jordan form with real eigenvalues $λ_{1}, \dots, λ_{r}$ of algebraic multiplicities $m_{1}, \dots, m_{r}$ summing to $n$ , every entry of $e^{- A^{T} t}$ is a sum $\sum_{j} q_{j} (t) e^{- λ_{j} t}$ with $de g q_{j} \leq m_{j} - 1$ . Thus $σ (t) = \sum_{j = 1}^{r} q_{j} (t) e^{- λ_{j} t}$ is a real exponential polynomial with $\sum_{j} (de g q_{j} + 1) \leq n$ terms. By the generalised Descartes / Pólya bound, a nonzero exponential polynomial $\sum_{j} q_{j} (t) e^{- λ_{j} t}$ with distinct real $λ_{j}$ has at most $(\sum_{j} (de g q_{j} + 1)) - 1 \leq n - 1$ real zeros. (One proves this by induction: factoring out $e^{- λ_{1} t}$ and differentiating $m_{1}$ times removes the first group and, by Rolle, drops the zero count by at most the group size minus the new degree gained; the count is bounded by total-terms-minus-one.) Controllability (normality) guarantees $σ \neq \equiv 0$ , so the zeros are finite in number and each is a switch. Hence the control switches at most $n - 1$ times. Rubric: full credit for the exponential-polynomial form, the term count $\leq n$ , the zero bound $\leq n - 1$ , and the normality appeal.

Exercise 8 (hard, symbolic).

Prove uniqueness of the time-optimal control for a normal single-input system: if $(A, b)$ is controllable and $u^{⋆}$ is time-optimal to a target, then $u^{⋆}$ is the unique time-optimal control, and it is bang-bang almost everywhere.

Hint

Two optimal controls share the same minimal time $T^{⋆}$ , so both reach the boundary of the attainable set $A (T^{⋆})$ at the target; use the separating-hyperplane costate and the isolated zeros of $σ$ .

Answer

The target $x_{T}$ at the minimal time $T^{⋆}$ lies on the boundary $\partial A (T^{⋆})$ of the attainable set (interior points are reachable in less than $T^{⋆}$ , contradicting minimality). The attainable set of a linear system with convex control set is convex and compact, so there is a supporting hyperplane at $x_{T}$ with outward normal $η \neq = 0$ ; setting $p (T^{⋆}) = η$ and solving $\overset{p}{˙} = - A^{T} p$ backward, every time-optimal control must maximise the boundary functional, i.e. minimise $σ (t)^{T} u$ pointwise, giving $u_{i}^{⋆} = - sign (σ_{i})$ wherever $σ_{i} \neq = 0$ . By normality (controllability of $(A, b)$ ), $σ (t) = b^{T} e^{- A^{T} t} η$ has only isolated zeros for each $η \neq = 0$ , so $u^{⋆}$ is determined almost everywhere by the single supporting normal $η$ — bang-bang with no singular arc. Any other time-optimal control would generate the same supporting hyperplane (the boundary point and the normal cone of the convex attainable set determine $η$ up to positive scaling, which does not change $sign (σ)$ ), hence the same control a.e. Therefore $u^{⋆}$ is unique. Rubric: full credit for the boundary/supporting-hyperplane setup, the costate from the normal, the bang-bang a.e. conclusion via isolated zeros, and the uniqueness from the determined normal.

Advanced results Master

Theorem (existence of time-optimal controls, Filippov; Hermes-LaSalle Ch. 2 ^{[source pending]}). Let $\overset{x}{˙} = A x + B u$ with $u (t) \in U$ , $U \subseteq R^{m}$ compact and convex, and suppose the target $x_{T}$ is reachable from $x_{0}$ by some admissible control. Then a time-optimal control exists. The argument is the compactness of the attainable set: for fixed $T$ the reachable set $A (T) = {x_{0} + \int_{0}^{T} e^{A (T - s)} B u (s) d s : u (\cdot) admissible}$ is convex (the control set is convex and the dynamics enter linearly) and compact (a continuous image of the weak- $*$ compact set of admissible controls), and $T \mapsto A (T)$ is continuous, so the infimal time at which $x_{T} \in A (T)$ is attained. Convexity of $U$ is what makes $A (T)$ closed; for nonconvex $U$ one must pass to the closed convex hull, which is the relaxation that the bang-bang principle then shows loses nothing.

Theorem (bang-bang principle via Lyapunov convexity; Hermes-LaSalle Ch. 3 ^{[source pending]}). The attainable set of bang-bang controls (valued at the vertices of the cube $U = [- 1, 1]^{m}$ ) equals the attainable set of all admissible controls valued in $U$ . Consequently every point reachable in time $T$ is reachable in time $T$ by a bang-bang control. The mechanism is Lyapunov's theorem on the range of a nonatomic vector-valued measure: the map $u (\cdot) \mapsto \int_{0}^{T} e^{A (T - s)} B u (s) d s$ has the same range over bang-bang controls as over all $U$ -valued controls because the range of a nonatomic vector measure is convex and compact, and its extreme points are images of extreme-valued (vertex) integrands. This is the deep reason the time-optimal control may be taken bang-bang even when uniqueness fails, and it is independent of the maximum principle: it is a measure-theoretic convexity statement about the reachable set.

Theorem (number of switches; Pontryagin et al. Ch. 3 ^{[source pending]}). For a normal single-input system $\overset{x}{˙} = A x + b u$ , $u \in [- 1, 1]$ , with $A$ having only real eigenvalues, the time-optimal control switches at most $n - 1$ times. If $A$ has a complex-conjugate pair of eigenvalues $α \pm i β$ , the switching function carries a factor $e^{α t} cos (β t - ϕ)$ whose zeros are infinite in number and spaced $π / β$ apart, so the switch count is unbounded as the horizon grows. The real-eigenvalue bound is the Pólya–Descartes zero bound for exponential polynomials applied to $σ (t) = b^{T} e^{- A^{T} t} p (0)$ , which has at most $n$ exponential-monomial terms and hence at most $n - 1$ real zeros. For the harmonic oscillator ( $n = 2$ , $α = 0$ , $β = 1$ ) the switching loci are translated half-circles in the phase plane, and the time-optimal synthesis is a sequence of semicircular arcs joined at switches spaced $π$ apart.

Theorem (normality and uniqueness; Athans-Falb Ch. 6 ^{[source pending]}; Hermes-LaSalle Ch. 4 ^{[source pending]}). If $(A, b_{i})$ is controllable for each input column $b_{i}$ (the system is normal), then for every reachable target the time-optimal control is unique and bang-bang almost everywhere, with no singular arcs. Conversely, loss of controllability of some $(A, b_{i})$ permits a switching component $σ_{i}$ to vanish on an interval, creating a singular arc on which the maximum principle does not determine $u_{i}$ . Normality is the precise hypothesis separating the clean bang-bang regime from the singular regime; on a singular arc the order of the singularity (the first time-derivative of $σ_{i}$ in which $u_{i}$ appears explicitly) and the generalised Legendre–Clebsch condition determine the singular control, a phenomenon outside the box-vertex picture.

Theorem (feedback synthesis for the double integrator; Athans-Falb Ch. 7 ^{[source pending]}). For $\overset{x}{˙}_{1} = x_{2}$ , $\overset{x}{˙}_{2} = u$ , $u \in [- 1, 1]$ , the time-optimal feedback to the origin is the state-feedback law

u^{⋆} (x) = ⎩ ⎨ ⎧ - 1, + 1, - sign (x_{2}), x_{1} + \frac{1}{2} x_{2} ∣ x_{2} ∣ > 0, x_{1} + \frac{1}{2} x_{2} ∣ x_{2} ∣ < 0, x_{1} + \frac{1}{2} x_{2} ∣ x_{2} ∣ = 0, x \neq = 0,

with $u^{⋆} (0) = 0$ . Above the switching curve the control brakes with $u = - 1$ until the state hits the curve, then rides the curve to the origin with $u = - sign (x_{2})$ ; below the curve the roles reverse. This converts the open-loop two-phase plan into a closed-loop law: the controller need not know the elapsed time, only the current state, and the switching curve is the locus where the bang flips. The discontinuity of $u^{⋆}$ across the curve is the synthesised image of the costate's sign change.

Synthesis. Minimum-time control is one principle — minimise the linear functional $σ (t)^{T} u$ over the control cube, which the maximum principle of 44.07.01 supplies — and every feature of the theory is a reading of that single fact. The central insight is that a linear functional on a polytope is minimised at a vertex, so the time-optimal control is bang-bang, $u_{i}^{⋆} = - sign (σ_{i})$ , with the switching function $σ = B^{T} p$ deciding the sign and the eigenstructure of $A$ deciding how often it flips. This is exactly why the double integrator switches once while the harmonic oscillator switches forever: $σ$ is affine for nilpotent $A$ and sinusoidal for an imaginary-eigenvalue $A$ , and the Pólya–Descartes bound on exponential-polynomial zeros is the foundational reason the real-eigenvalue switch count is capped at $n - 1$ . The bang-bang principle generalises beyond the necessary conditions through Lyapunov's convexity theorem, which shows the reachable set of vertex-valued controls already fills the reachable set of all controls — putting these together, existence (Filippov, by compactness of the reachable set), the bang-bang property, and uniqueness (by normality and the supporting-hyperplane costate) are three faces of the convex geometry of the attainable set. The construction is dual to the linear-quadratic regulator of 44.07.02: quadratic cost makes $H$ strictly convex in $u$ with an interior smooth minimiser and continuous feedback $u = - K x$ , while time cost makes $H$ linear with a vertex minimiser and discontinuous switching-curve feedback, and the bridge is the same Hamiltonian/costate machinery specialised to two running costs.

Full proof set Master

Proposition 1 (bang-bang from linearity). For $\overset{x}{˙} = A x + B u$ , $u \in [- 1, 1]^{m}$ , minimum time, the optimal control satisfies $u_{i}^{⋆} (t) = - sign (σ_{i} (t))$ with $σ = B^{T} p$ , $\overset{p}{˙} = - A^{T} p$ , wherever $σ_{i} (t) \neq = 0$ .

Proof. This is the Key theorem. The maximum principle minimises $H = 1 + p^{T} A x + σ^{T} u$ over $u \in [- 1, 1]^{m}$ ; the $u$ -dependence is the separable linear form $\sum_{i} σ_{i} u_{i}$ , each term minimised over $[- 1, 1]$ at $u_{i} = - sign (σ_{i})$ when $σ_{i} \neq = 0$ . The adjoint $\overset{p}{˙} = - A^{T} p$ gives $p (t) = e^{- A^{T} t} p (0)$ , so $σ (t) = B^{T} e^{- A^{T} t} p (0)$ . $□$

Proposition 2 (affine switching function for the double integrator). For $\overset{x}{˙}_{1} = x_{2}$ , $\overset{x}{˙}_{2} = u$ , the switching function is $σ (t) = c_{2} - c_{1} t$ for constants $c_{1}, c_{2}$ , so there is at most one switch.

Proof. $A = \begin{psmallmatrix} 0 & 1 \\ 0 & 0\end{psmallmatrix}$ , $b = \begin{psmallmatrix} 0 \\ 1\end{psmallmatrix}$ . The adjoint $\overset{p}{˙} = - A^{T} p$ gives $\overset{p}{˙}_{1} = 0$ , $\overset{p}{˙}_{2} = - p_{1}$ , so $p_{1} = c_{1}$ and $p_{2} = c_{2} - c_{1} t$ . Then $σ = b^{T} p = p_{2} = c_{2} - c_{1} t$ , affine, with at most one zero unless $c_{1} = c_{2} = 0$ , excluded by the nontriviality of the costate under controllability of $(A, b)$ . $□$

Proposition 3 (the switching curve). The single-final-arc loci reaching the origin at rest for the double integrator form the curve $γ = {x_{1} + \frac{1}{2} x_{2} ∣ x_{2} ∣ = 0}$ .

Proof. Integrate $u \equiv \pm 1$ backward from the origin: $x_{2} (- s) = - u s$ , $x_{1} (- s) = \frac{1}{2} u s^{2}$ for $s \geq 0$ . From $x_{2} = - u s$ and $u^{2} = 1$ , $s = - u x_{2} \geq 0$ requires $u = - sign (x_{2})$ , and $x_{1} = \frac{1}{2} u x_{2}^{2} = - \frac{1}{2} sign (x_{2}) x_{2}^{2} = - \frac{1}{2} x_{2} ∣ x_{2} ∣$ . The union over both signs is $γ$ . $□$

Proposition 4 (number of switches under real eigenvalues). If $A$ has only real eigenvalues and $(A, b)$ is controllable, the time-optimal control switches at most $n - 1$ times.

Proof. $σ (t) = b^{T} e^{- A^{T} t} p (0) = \sum_{j = 1}^{r} q_{j} (t) e^{- λ_{j} t}$ with distinct real $λ_{j}$ and $\sum_{j} (de g q_{j} + 1) \leq n$ . A nonzero real exponential polynomial $\sum_{j} q_{j} (t) e^{- λ_{j} t}$ has at most $(\sum_{j} (de g q_{j} + 1)) - 1 \leq n - 1$ real zeros: argue by induction on the number of terms $N = \sum_{j} (de g q_{j} + 1)$ . For $N = 1$ , $q_{1} (t) e^{- λ_{1} t}$ has at most $de g q_{1} = N - 1$ zeros. For the step, divide by $e^{- λ_{1} t}$ (no change in zeros) and differentiate $de g q_{1} + 1$ times; this annihilates the first group and yields an exponential polynomial with $N - (de g q_{1} + 1)$ terms whose zero count, by the inductive hypothesis and Rolle's theorem (each differentiation reduces the zero count by at most one and is bounded below by the next stage's count), gives the bound. Controllability makes $σ \neq \equiv 0$ , so the zeros are finite and each is a switch. $□$

Proposition 5 (uniqueness under normality). For a normal single-input system, the time-optimal control to a reachable target is unique and bang-bang a.e.

Proof. At the minimal time $T^{⋆}$ the target lies on $\partial A (T^{⋆})$ , the boundary of the convex compact attainable set. A supporting hyperplane at the target has outward normal $η \neq = 0$ , unique up to positive scaling since the target is an exposed boundary point of a convex set reached at minimal time. Setting $p (T^{⋆}) = η$ and integrating $\overset{p}{˙} = - A^{T} p$ backward, every time-optimal control minimises $σ (t)^{T} u$ pointwise, so $u^{⋆} = - sign (σ)$ where $σ \neq = 0$ . Normality (controllability of $(A, b)$ ) makes $σ (t) = b^{T} e^{- A^{T} t} η$ have isolated zeros for every $η \neq = 0$ , so $u^{⋆}$ is determined a.e. by $η$ ; positive rescaling of $η$ leaves $sign (σ)$ unchanged, so the control is the same for every supporting normal. Hence the time-optimal control is unique and bang-bang a.e. $□$

Connections Master

The bang-bang principle is the boundary-minimisation face of Pontryagin's maximum principle of 44.07.01: where the general principle states the optimal control minimises the Hamiltonian pointwise over the constraint set, the linear-in-control minimum-time problem makes that minimisation a linear functional on a cube, so the optimum is a vertex and the control is $u_{i}^{⋆} = - sign (σ_{i})$ with $σ = B^{T} p$ the costate-projected coefficient, the explicit instance the maximum principle was built to handle.

Minimum-time control is the discontinuous-feedback counterpart of the linear-quadratic regulator of 44.07.02: both ride the same costate dynamics $\overset{p}{˙} = - A^{T} p$ and the same Hamiltonian structure, but the quadratic cost of LQR makes $H$ strictly convex in $u$ with an interior minimiser and continuous gain $u = - K x$ , while the time cost makes $H$ linear in $u$ with a vertex minimiser and a discontinuous switching-curve feedback, so the two units are the convex and the linear specialisations of one Hamiltonian framework.

The switching function and the switch-count bound rest on the matrix-exponential and reachability theory of 02.06.03: the costate solves $p (t) = e^{- A^{T} t} p (0)$ , the switching function $σ = B^{T} e^{- A^{T} t} p (0)$ is an exponential polynomial whose zero structure is set by the spectrum of $A$ , and normality is exactly the Kalman controllability rank condition on $[b A b \dots A^{n - 1} b]$ that guarantees isolated switches and a controllable, hence reachable, target.

The existence and bang-bang theorems are convex-geometry statements about the attainable set, linking forward to convex analysis (44.01.01): Filippov existence is the compactness of the reachable set under a convex control constraint, and the bang-bang principle is Lyapunov's theorem that a nonatomic vector measure has convex range with vertex-valued extreme points, so the reachable set of bang-bang controls already equals the reachable set of all controls.

The singular-arc phenomenon forward-links to the broader optimal-control theory of singular controls and the Hamilton-Jacobi-Bellman value function of 44.07.04: when normality fails and $σ_{i} \equiv 0$ on an interval the maximum principle underdetermines the control, and the HJB value function's nonsmoothness across the switching surface is the dynamic-programming image of the bang-bang discontinuity, with the switching curve appearing as the locus where the value gradient $p = \partial_{x} V$ changes the sign of $σ$ .

Historical & philosophical context Master

The bang-bang principle emerged directly from the work of Lev Pontryagin and his collaborators Vladimir Boltyanskii, Revaz Gamkrelidze, and Evgenii Mishchenko at the Steklov Institute in the mid-1950s; the time-optimal control of bounded linear systems was the motivating class of problems, drawn from aircraft and missile guidance, that exposed the inadequacy of the classical calculus of variations and prompted the maximum principle ^{[Pontryagin 1962]}. The recognition that the optimal control jumps between extreme values, and that for the double integrator there is a single switch governed by a parabolic switching curve, is worked out in their monograph alongside the harmonic-oscillator example with its periodic switching.

The measure-theoretic foundation — that bang-bang controls reach everything all admissible controls reach — was placed on rigorous footing by Aleksei Lyapunov's 1940 theorem on the convexity of the range of a nonatomic vector measure, and its application to control existence and the bang-bang property was developed by LaSalle, Hermes, and Aleksei Filippov around 1960; Filippov's existence theorem, resting on the closedness of the reachable set under a convex control constraint, became the standard existence tool ^{[Filippov 1962]}. The synthesis problem — converting the open-loop optimal control into a state-feedback switching law — and the systematic treatment of the switching function, normality, and the number of switches were consolidated by Michael Athans and Peter Falb in their 1966 text, which remains the canonical engineering reference for time-optimal design ^{[Athans 1966]}.

Bibliography Master

@book{Pontryagin1962BangBang,
  author    = {Pontryagin, Lev S. and Boltyanskii, Vladimir G. and Gamkrelidze, Revaz V. and Mishchenko, Evgenii F.},
  title     = {The Mathematical Theory of Optimal Processes},
  publisher = {Interscience Publishers (Wiley)},
  address   = {New York},
  year      = {1962},
  note      = {English translation by K. N. Trirogoff of the 1961 Russian original}
}

@book{AthansFalb1966BangBang,
  author    = {Athans, Michael and Falb, Peter L.},
  title     = {Optimal Control: An Introduction to the Theory and Its Applications},
  publisher = {McGraw-Hill},
  address   = {New York},
  year      = {1966}
}

@book{HermesLaSalle1969,
  author    = {Hermes, Henry and LaSalle, Joseph P.},
  title     = {Functional Analysis and Time Optimal Control},
  publisher = {Academic Press},
  address   = {New York},
  year      = {1969}
}

@book{Liberzon2012BangBang,
  author    = {Liberzon, Daniel},
  title     = {Calculus of Variations and Optimal Control Theory: A Concise Introduction},
  publisher = {Princeton University Press},
  address   = {Princeton, NJ},
  year      = {2012}
}

@article{Filippov1962,
  author  = {Filippov, Aleksei F.},
  title   = {On Certain Questions in the Theory of Optimal Control},
  journal = {SIAM Journal on Control},
  volume  = {1},
  number  = {1},
  year    = {1962},
  pages   = {76--84},
  note    = {English translation of the 1959 Russian original}
}

@article{LaSalle1960,
  author  = {LaSalle, Joseph P.},
  title   = {The Time Optimal Control Problem},
  journal = {Contributions to the Theory of Nonlinear Oscillations},
  volume  = {5},
  publisher = {Princeton University Press},
  year    = {1960},
  pages   = {1--24}
}

@article{Lyapunov1940,
  author  = {Lyapunov, Aleksei A.},
  title   = {Sur les fonctions-vecteurs compl\`etement additives},
  journal = {Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya},
  volume  = {4},
  year    = {1940},
  pages   = {465--478}
}

@book{LeeMarkus1967,
  author    = {Lee, Ernest B. and Markus, Lawrence},
  title     = {Foundations of Optimal Control Theory},
  publisher = {Wiley},
  address   = {New York},
  year      = {1967}
}

Prerequisites

44.07.01
44.07.02
02.06.03

Tier anchors

beginner: Driving a cart to a parking spot in the least time when the pedal can only be floored forward or floored back, so the whole plan is 'slam one way, then slam the other once' — Liberzon 2012 Calculus of Variations and Optimal Control Theory: A Concise Introduction (Princeton University Press) Ch. 4 (the bang-bang principle and the time-optimal double integrator); Athans & Falb 1966 Optimal Control: An Introduction to the Theory and Its Applications (McGraw-Hill) Ch. 7 (minimum-time problems and the switching curve)
intermediate: Liberzon 2012 Calculus of Variations and Optimal Control Theory (Princeton University Press) Ch. 4 (the switching function, the bang-bang principle for linear systems, the time-optimal double integrator and its switching curve); Athans & Falb 1966 Optimal Control (McGraw-Hill) Ch. 6-7 (the minimum-time problem for $\overset{x}{˙} = A x + B u$ , the switching function $σ = B^{T} p$ , normality and uniqueness, the number-of-switches bound)
master: Pontryagin, Boltyanskii, Gamkrelidze & Mishchenko 1962 The Mathematical Theory of Optimal Processes (Interscience/Wiley) Ch. 2-3 (the maximum principle for time-optimal linear systems, the bang-bang theorem, the number of switches bounded by the eigenvalue structure, the harmonic-oscillator example); Hermes & LaSalle 1969 Functional Analysis and Time Optimal Control (Academic Press) Ch. 2-4 (existence via Filippov, the bang-bang principle as an extreme-point/Lyapunov-convexity statement, normality and uniqueness); Athans & Falb 1966 Optimal Control (McGraw-Hill) Ch. 7 (synthesis of the time-optimal feedback and the switching curve)

References

Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V. & Mishchenko, E. F. — The Mathematical Theory of Optimal Processes (Interscience/Wiley, 1962; English translation of the 1961 Russian original) · Ch. 2-3: the maximum principle specialised to the time-optimal problem for linear systems dx/dt = Ax + Bu with control constrained to a cube |u_i| <= 1; the switching function as the costate-projected control coefficient B^T p; the bang-bang theorem that an optimal control takes values at the vertices of the control cube; the number of switches of each control component bounded by n-1 when A has only real eigenvalues; normality and the uniqueness of the time-optimal control; the harmonic-oscillator example with its arc-of-circle switching loci and the double-integrator example with its parabolic switching curve.
Athans, M. & Falb, P. L. — Optimal Control: An Introduction to the Theory and Its Applications (McGraw-Hill, 1966) · Ch. 6-7: the minimum-time problem min T subject to dx/dt = Ax + Bu, |u_i| <= 1, fixed endpoints; the Hamiltonian H = 1 + p^T(Ax + Bu), the costate equation dp/dt = -A^T p, and the switching function sigma(t) = B^T p(t); the bang-bang principle u_i^* = -sign(sigma_i); normality of (A,b) as the condition that sigma_i has only isolated zeros; the double-integrator time-optimal solution and its parabolic switching curve x_1 + (1/2) x_2 |x_2| = 0; synthesis of the time-optimal feedback law.
Hermes, H. & LaSalle, J. P. — Functional Analysis and Time Optimal Control (Academic Press, 1969) · Ch. 2-4: existence of time-optimal controls via the Filippov / compactness-of-the-attainable-set argument; the bang-bang principle as a statement that the attainable set of bang-bang controls equals the attainable set of all admissible controls, via Lyapunov's theorem on the range of a nonatomic vector measure; normality and uniqueness of the time-optimal control; the relation between the switching function and the support functional of the reachable set.

Estimated time

beginner: 18m
intermediate: 48m
master: 90m