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Introduction The maximum principle and its consequences Approximation by finite dimensional systems Some recent results and open questions in time optimal 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 Introduction 1 The maximum principle and its consequences 2 The case of point target The case of a ball target Approximation by finite dimensional systems 3 Marius Tucsnak Time optimal control

Introduction The maximum principle and its consequences Approximation by finite dimensional systems Outline Introduction 1 The maximum principle and its consequences 2 The case of point target The case of a ball target Approximation by finite dimensional systems 3 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 z 0 to a target B ( z 1 , 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 L 2 in time control u steering z 0 to B ( z 1 , ε ) ? (reachability, controllability) Is there any L ∞ in time control with � u ( t ) � ≤ 1 steering z 0 to B ( z 1 , 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 z 0 to the final position z 1 , with a bounded acceleration: z ( t ) = u ( t ) , ¨ z (0) = z 0 , z (0) = 0 , ˙ z ( τ ) = z 1 , 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 on Γ × (0 , ∞ ) , − 1 � u � 1 z = 0 on ( ∂ Ω \ Γ) × (0 , ∞ ) , z ( x, 0) = z 0 ( 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 = C n and U = C m for finite-dimensional control systems. T = ( T t ) t � 0 is a strongly continuous semigroup on X generated by A . We have T t = e tA for for finite-dimensional control systems. B ∈ L ( U ; X ) be a control operator and let u ∈ L 2 ([0 , ∞ ) , U ) be an input function. We have B ∈ L ( C m ; C n ) 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 ) = T t z (0) + Φ t u , where � t Φ t ∈ L ( L 2 ([0 , ∞ ) , U ); X ) , Φ t u = 0 T t − σ Bu ( σ ) d σ. ( A, B ) is said exactly controllable in time τ if Ran Φ τ = X . ( A, B ) is said null controllable in time τ if Ran Φ τ ⊃ Ran T τ . This is equivalent to the existence of K τ > 0 such that � τ t ϕ � 2 d t � � T ∗ τ ϕ � 2 (final-state observability) K 2 0 � B ∗ T ∗ τ ( A, B ) is approximatively controllable in time τ if Ran Φ τ = X , or, ( 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) U ad = { u ∈ L ∞ ([0 , ∞ ) , U ) | � u ( t ) � � 1 a. e. in [0 , ∞ ) } . Assume that z 0 , z 1 ∈ X are s.t. there exists u ∈ U ad and τ > 0 s.t. z 1 = T τ z 0 + Φ τ u ( z 1 reachable from z 0 ). Determine τ ∗ ( z 0 , z 1 ) = min { τ | z 1 = T τ z 0 + Φ τ u } . u ∈ U ad 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 optimal 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 ole 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 case of point target The maximum principle and its consequences The case of a ball target Approximation by finite dimensional systems Outline Introduction 1 The maximum principle and its consequences 2 The case of point target The case of a ball target Approximation by finite dimensional systems 3 Marius Tucsnak Time optimal control

Introduction The case of point target The maximum principle and its consequences The case of a ball target Approximation by finite dimensional systems Finite dimensional linear systems Assume that X = C n , U = C m . 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 ∗ − t z, u ∗ ( t ) � = max � u �≤ 1 Re � B ∗ T ∗ τ ∗ − t z, u � Corollary 1 If ( A, B ) controllable then the time optimal control u ∗ is bang-bang. More precisely, u ∗ is uniquely determined by 1 u ∗ ( t ) = τ ∗ − t z � B ∗ T ∗ ( t ∈ [0 , τ ∗ ] τ ∗ − t z a.e. ) . � B ∗ T ∗ Marius Tucsnak Time optimal control

Introduction The case of point target The maximum principle and its consequences The case of a ball target Approximation by finite dimensional systems 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 on [0 , τ ∗ ] . Then there exists z ∈ X , z � = 0 such that Re � B ∗ T ∗ τ ∗ − t z, u ∗ ( t ) � = max � u �≤ 1 Re � B ∗ T ∗ τ ∗ − t z, u � Moreover, assume that ( A, B ) is approximatively controllable from sets of positive measure. Then u ∗ ( t ) is bang-bang, unique and given by 1 u ∗ ( t ) = τ ∗ − t z � B ∗ T ∗ ( t ∈ [0 , τ ∗ ] τ ∗ − t z a.e. ) , � B ∗ T ∗ Marius Tucsnak Time optimal control