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Introduction The maximum principle and its consequences Approximation by finite dimensional systems

Some recent results and open questions in time

• ptimal control for infinite dimensional systems

Marius Tucsnak

Universit´ e de Lorraine

Toulouse, June 2014

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems

Outline

1

Introduction

2

The maximum principle and its consequences The case of point target The case of a ball target

3

Approximation by finite dimensional systems

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems

Outline

1

Introduction

2

The maximum principle and its consequences The case of point target The case of a ball target

3

Approximation by finite dimensional systems

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems

Problem statement

General Aim: Steer a given system from a given initial state z0 to a target B(z1, ε), using a control function u∗ such that We have u∗(t) 1 for almost every time t The control u∗ is doing the job in a minimal time Questions to be addressed: Is there any L2 in time control u steering z0 to B(z1, ε) ? (reachability, controllability) Is there any L∞ in time control with u(t) ≤ 1 steering z0 to B(z1, ε) ? ( constrained reachability) Existence of a time optimal control. Is the time optimal control a bang-bang one, i.e., do we have u∗(t) = 1 a.e.? Computation of the time optimal control

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems

First examples(I)

Steer the rocket car from rest to rest, from the initial position z0 to the final position z1, with a bounded acceleration: ¨ z(t) = u(t), z(0) = z0, ˙ z(0) = 0, z(τ) = z1, ˙ z(τ) = 0, −1 u(t) 1. The solution is to apply max acceleration, i.e., u∗(t) = 1 until the unique switching point, and then max braking, i.e., u∗(t) = −1. This solution, can be “uncomfortable” for the passengers...

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems

First examples(II)

Controlling temperature in a room by acting on the boundary: ˙ z = ∆z in Ω × (0, ∞) z = u

• n Γ × (0, ∞),

−1 u 1 z = 0

• n

(∂Ω \ Γ) × (0, ∞), z(x, 0) = z0(x) for x ∈ Ω, z(x, T) = 0 for x ∈ Ω We can prove that the minimal time control satisfies |u(x, t)| = 1 a.e. (S.Micu,I.Roventa and

• M. T, JFA, 2012)

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems

The linear case (I): Notation

X (the state space) and U (the input space) are complex Hilbert spaces. We have X = Cn and U = Cm for finite-dimensional control systems. T = (Tt)t0 is a strongly continuous semigroup on X generated by A. We have Tt = etA for for finite-dimensional control systems. B ∈ L(U; X) be a control operator and let u ∈ L2([0, ∞), U) be an input function. We have B ∈ L(Cm; Cn) for finite-dimensional systems.

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems

The linear case (II): some background

Let ˙ z(t) = Az(t) + Bu(t), or z(t) = Ttz(0) + Φtu, where Φt ∈ L(L2([0, ∞), U); X), Φtu = t

0 Tt−σBu(σ) dσ.

(A, B) is said exactly controllable in time τ if Ran Φτ = X. (A, B) is said null controllable in time τ if Ran Φτ ⊃ Ran Tτ. This is equivalent to the existence of Kτ > 0 such that K2

τ

τ

0 B∗T∗ t ϕ2 dt T∗ τϕ2 (final-state observability)

(A, B) is approximatively controllable in time τ if Ran Φτ = X,

• r, (B∗T∗

t ϕ = 0 for t ∈ [0, τ] ⇒ ϕ = 0) (approx. observability)

In the last 30 years, the above properties have been investigated for a large number of PDE’s

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems

The commercial break:

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems

The linear case (III): precise statement (with point target)

Uad = {u ∈ L∞([0, ∞), U) | u(t) 1 a. e. in [0, ∞)}. Assume that z0, z1 ∈ X are s.t. there exists u ∈ Uad and τ > 0 s.t. z1 = Tτz0 + Φτu (z1 reachable from z0). Determine τ ∗(z0, z1) = min

u∈Uad

{τ | z1 = Tτz0 + Φτu}. Determine the properties (namely the bang-bang one) of the corresponding control u∗. Give methods to compute this control. Why such a basic control question is studied only recently in the PDE’s case? The foundations and some of the main results of infinite dimensional systems theory are relatively recent The passage from “classical controllability” to time optimal control requires new (untrivial) results

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems

Bibliography(I)

Fattorini, H. O.: Time-optimal control of solutions of operational differential equations, J. Soc. Indust. Appl. Math. Control, 1964. Fattorini, H. O.: A remark on the ”bang-bang” principle for linear control systems in infinite-dimensional space, SIAM J. Control, 1968. Schmidt, G.: The ”bang-bang” principle for the time-optimal problem in boundary control of the heat equation, SIAM J. Control Optim., 1980. Mizel, V. J. and Seidman, T. I.: An abstract bang-bang principle and time-optimal boundary control of the heat equation, SIAM J. Control Optim., 1997. Wang, G. and Wang, L., The bang-bang principle of time optimal controls for the heat equation with internal controls, Systems Control Lett., 2007.

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems

Bibliography (II)

Phung, K. D., Wang, G., Zhang, X.: On the existence of time

• ptimal controls for linear evolution equations, Discrete Contin.
• Dyn. Syst. Ser. B, 2007.

Wang, G.: L∞-null controllability for the heat equation and its consequences for the time optimal control problem, SIAM J. Control Optim., 2008. Phung, K. D., Wang, G. An observability for parabolic equations from a measurable set in time, JEMS 15 (2013), 681–703.

Books:

Lions J.-L.: Contrˆ

• le optimal de syst`

emes gouvern´ es par des ´ equations aux d´ eriv´ ees partielles, Gauthier-Villars, Paris, 1968. Fattorini, H. O.: Infinite dimensional linear control systems. The time optimal and norm optimal problems, North-Holland Mathematics Studies, 201, Amsterdam, 2005.

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems The case of point target The case of a ball target

Outline

1

Introduction

2

The maximum principle and its consequences The case of point target The case of a ball target

3

Approximation by finite dimensional systems

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems The case of point target The case of a ball target

Finite dimensional linear systems

Assume that X = Cn, U = Cm. Theorem 1 (Maximum Principle, Bellman et al. (1956)) Let u∗(t) be the time optimal control, defined on [0, τ ∗]. Then there exists z ∈ X, z = 0 such that Re B∗T∗

τ ∗−tz, u∗(t) = max u≤1 Re B∗T∗ τ ∗−tz, u

Corollary 1 If (A, B) controllable then the time optimal control u∗ is bang-bang. More precisely, u∗ is uniquely determined by u∗(t) = 1 B∗T∗

τ ∗−tzB∗T∗ τ ∗−tz

(t ∈ [0, τ ∗] a.e.).

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems The case of point target The case of a ball target

An infinite dimensional extension

Theorem 1 (J. Loh´ eac and M.T., SICON 2013) Assume that B ∈ L(U, X) and that (A, B) is exactly controllable in any time τ > 0. Let u∗(t) be the time optimal control, defined

• n [0, τ ∗]. Then there exists z ∈ X, z = 0 such that

Re B∗T∗

τ ∗−tz, u∗(t) = max u≤1 Re B∗T∗ τ ∗−tz, u

Moreover, assume that (A, B) is approximatively controllable from sets of positive measure. Then u∗(t) is bang-bang, unique and given by u∗(t) = 1 B∗T∗

τ ∗−tzB∗T∗ τ ∗−tz

(t ∈ [0, τ ∗] a.e.),

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems The case of point target The case of a ball target

Idea of the proof

For each τ > 0, we endow X with the norm |||z||| = inf{uL∞([0,τ],U) | Φτu = z}. Note that ||| · ||| is equivalent with the original norm · . For τ > 0 we set B∞(τ) = {Φτu | uL∞([0,τ],U) 1}, and we show that if (τ ∗, u∗) is an optimal pair then Φτ ∗u∗ ∈ ∂B∞(τ ∗). Using the fact that B∞(τ ∗) has a non empty interior, we apply a geometric version of the Hahn-Banach theorem to get the conclusion.

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems The case of point target The case of a ball target

An infinite dimensional extension

Theorem 2 (Maximum Principle) Assume that the target is B(0, ε). Assume that B ∈ L(U, X) and that (A, B) generates a semigroup T. Then there exists ξ ∈ X, ξ = 0 such that, for almost every σ ∈ [0, τ ∗],

• u∗(σ), B∗T∗

τ ∗−σξ

• U =

max

v∈U, v1

• v, B∗T∗

τ ∗−σξ

• U .

Moreover, ξ satisfies the transversality condition, i.e. : ξ = − (Tτ ∗z0 + Φτ ∗u∗) .

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems The case of point target The case of a ball target

Sketch of the proof (Chi-Ting Wu, M.T. and J. Valein)

Let z1 = Tτ ∗z0 + Φτ ∗u∗ be the “entrance” point. Denote Uτ := {u|[0,τ] | u ∈ Uad}, and Rτz0 = Tτz0 + ΦτUτ. First step. Show that z1 ∈ ∂Rτ ∗z0 ∩ ∂B(0, ε). Second step. Use the Hahn-Banach theorem to show the existence

• f ξ = 0 and α ∈ R with ξ, η α for η ∈ Rτ ∗z0, and ξ, η α

for η ∈ B(0, ε)). Third step. Combine the first two steps to get ξ, z1 ξ, η (η ∈ Rτ ∗z0), which implies the conclusions.

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems

Outline

1

Introduction

2

The maximum principle and its consequences The case of point target The case of a ball target

3

Approximation by finite dimensional systems

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems

Leading assumptions and notation

A = −A0 with A0 strictly positive. Set Xγ = D(Aγ

0) and

xγ = Aγ

0x.

(H1) For every h ∈ (0, h1) and 0 γ 1 we have : x − Phx Chθ1γxγ (x ∈ Xγ), where Ph is the orthogonal projector from X onto the approximation space Vh. (H2) Denote U = B∗X 1

2 , Uh = B∗Vh. We assume that

Qhu − uU ≤ Chθ2uV , where Qh is the projector from U onto Uh.

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems

Convergence results

Denote by (τ ∗

h, u∗ h) the solutions of the time optimal control

problem ˙ zh(t) = Ahzh(t) + Bhuh(t) (t ≥ 0), zh(0) = z0,h = Phz0. Theorem 3 Assume that z0 ∈ X. Then we have τ ∗

h(Phz0) → τ ∗(z0).

Moreover, u∗

h → u∗ in L2(0, τ ∗; U).

Remark 1 The maximum principle is used for the convergence of controls.

Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems

Some open questions

Bang-bang property for Euler-Bernoulli type equations L∞ boundary controls for Schr¨

• dinger type equations

Numerical analysis for more general targets and/or equations Error estimates.

Marius Tucsnak Time optimal control