Time optimal control problems: Bang-bang property and observability estimates from measurable sets Can ZHANG (zhangcansx@163.com) Paris, June 2016 In honor of Prof. J. M. Coron’s 60th birthday occasion C. ZHANG Time optimal control problems 1 / 21
Motivations 1 Observability estimates over measurable sets implies the bang-bang property Main results 2 Conclusion and further research 3 C. ZHANG Time optimal control problems 2 / 21
Bang-bang property of time optimal controls We introduce the time optimal control problem for the heat equation. Let ω ⊂ Ω be an open subset with its characteristic function χ ω . Consider the time optimal control problem: ( TP ) M T ( M ) � inf � � 1 : t > 0 : y ( t ; u ) = 0 , u ∈U M C. ZHANG Time optimal control problems 2 / 21
Bang-bang property of time optimal controls We introduce the time optimal control problem for the heat equation. Let ω ⊂ Ω be an open subset with its characteristic function χ ω . Consider the time optimal control problem: ( TP ) M T ( M ) � inf � � 1 : t > 0 : y ( t ; u ) = 0 , u ∈U M where U M is the control constraint given by U M = u ∈ L ∞ (Ω × R + ) : | u ( x , t ) | ≤ M for a.e. ( x , t ) ∈ Ω × R + � � , M > 0 , and y ( · ; u ) solves the non-homogeneous heat equation in Ω × R + , y t − ∆ y = χ ω u on ∂ Ω × R + , y = 0 y ( x , 0) = y 0 ( x ) in Ω . C. ZHANG Time optimal control problems 2 / 21
• We call T ( M ) the optimal time (if it exists) and u ∗ a time optimal control if y ( T ( M ); u ∗ ) = 0. C. ZHANG Time optimal control problems 3 / 21
• We call T ( M ) the optimal time (if it exists) and u ∗ a time optimal control if y ( T ( M ); u ∗ ) = 0. In fact, the existence of optimal time and time optimal controls are guaranteed by the observability inequality and the decay of energy of the heat equation. C. ZHANG Time optimal control problems 3 / 21
• We call T ( M ) the optimal time (if it exists) and u ∗ a time optimal control if y ( T ( M ); u ∗ ) = 0. In fact, the existence of optimal time and time optimal controls are guaranteed by the observability inequality and the decay of energy of the heat equation. • In the state space L 2 (Ω), since the reachable set � y ( T ( M ); u ) : u ∈ U M � has no interior point in L 2 (Ω), to the best of our knowledge, we do not know how to separate this set from the target set { 0 } by a hyperplane in L 2 (Ω). Thus, we do not know how to get the Pontryagin maximum principle for Problem ( TP ) M 1 by the way used in the case of O.D.E.. C. ZHANG Time optimal control problems 3 / 21
• We call T ( M ) the optimal time (if it exists) and u ∗ a time optimal control if y ( T ( M ); u ∗ ) = 0. In fact, the existence of optimal time and time optimal controls are guaranteed by the observability inequality and the decay of energy of the heat equation. • In the state space L 2 (Ω), since the reachable set � y ( T ( M ); u ) : u ∈ U M � has no interior point in L 2 (Ω), to the best of our knowledge, we do not know how to separate this set from the target set { 0 } by a hyperplane in L 2 (Ω). Thus, we do not know how to get the Pontryagin maximum principle for Problem ( TP ) M 1 by the way used in the case of O.D.E.. • It is natural to ask if the bang-bang property (for simplicity B-B-P) holds 1 : Any time optimal control u ∗ for ( TP ) M for ( TP ) M 1 verifies | u ∗ ( x , t ) | = M for a.e. ( x , t ) ∈ ω × (0 , T ( M )). C. ZHANG Time optimal control problems 3 / 21
The advantage of B-B-P • The time optimal control is unique. The argument is very simple. Assume u ∗ and v ∗ are two time optimal controls. Since = | u ∗ ( x , t ) | 2 + | v ∗ ( x , t ) | 2 � u ∗ ( x , t ) − v ∗ ( x , t ) � u ∗ ( x , t ) + v ∗ ( x , t ) 2 2 � � � � − , � � � � 2 2 2 � � and ( u ∗ + v ∗ ) / 2 is also a time optimal control, by the B-B-P, we get that u ∗ = v ∗ a.e. in ω × (0 , T ( M )). C. ZHANG Time optimal control problems 4 / 21
The advantage of B-B-P • The time optimal control is unique. The argument is very simple. Assume u ∗ and v ∗ are two time optimal controls. Since = | u ∗ ( x , t ) | 2 + | v ∗ ( x , t ) | 2 � u ∗ ( x , t ) − v ∗ ( x , t ) � u ∗ ( x , t ) + v ∗ ( x , t ) 2 2 � � � � − , � � � � 2 2 2 � � and ( u ∗ + v ∗ ) / 2 is also a time optimal control, by the B-B-P, we get that u ∗ = v ∗ a.e. in ω × (0 , T ( M )). • The equivalence between the time optimal control problem and norm optimal control problem, and so provide a necessary and sufficient condition for the time optimal control problem. G. Wang, E. Zuazua, On the equivalence of minimal time and minimal norm controls for internally controlled heat equations, SICON, 2012. C. ZHANG Time optimal control problems 4 / 21
Observability inequalities from measurable sets To present our motivations, we begin with the simplest situation. Let T > 0 and Ω be a bounded Lipschitz domain in R n . Consider the heat equation ∂ t u − ∆ u = 0 , in Ω × (0 , T ) , u = 0 , on ∂ Ω × (0 , T ) , u ( · , 0) = u 0 ∈ L 2 (Ω) . Two important a priori estimates for the above equation are as follows. Interior case: � | u ( x , t ) | dxdt , ∀ u 0 ∈ L 2 (Ω) , � u ( T ) � L 2 (Ω) ≤ N (Ω , T , D ) (1) D where D is a measurable subset of Ω × (0 , T ). C. ZHANG Time optimal control problems 5 / 21
Boundary case: � | ∂ ∂ν u ( x , t ) | d σ dt , ∀ u 0 ∈ L 2 (Ω) , � u ( T ) � L 2 (Ω) ≤ N (Ω , T , J ) (2) J where J is a measurable subset of ∂ Ω × (0 , T ). C. ZHANG Time optimal control problems 6 / 21
Boundary case: � | ∂ ∂ν u ( x , t ) | d σ dt , ∀ u 0 ∈ L 2 (Ω) , � u ( T ) � L 2 (Ω) ≤ N (Ω , T , J ) (2) J where J is a measurable subset of ∂ Ω × (0 , T ). • Such a priori estimates are usually called observability inequalities in Control Theory when the observation regions are open subsets. C. ZHANG Time optimal control problems 6 / 21
Boundary case: � | ∂ ∂ν u ( x , t ) | d σ dt , ∀ u 0 ∈ L 2 (Ω) , � u ( T ) � L 2 (Ω) ≤ N (Ω , T , J ) (2) J where J is a measurable subset of ∂ Ω × (0 , T ). • Such a priori estimates are usually called observability inequalities in Control Theory when the observation regions are open subsets. Our aim is to build up estimates (1) and (2) when D and J are subsets of positive measure and positive surface measure, respectively. C. ZHANG Time optimal control problems 6 / 21
Observability estimates from measurable sets implies the B-B-P Main idea : By contradiction, we would suppose that there were a constant ε ∈ (0 , M ) and a subset of positive measure D ⊂ ω × (0 , T ( M )) such that | u ∗ ( x , t ) | ≤ M − ε, ∀ ( x , t ) ∈ D . It provides a “room” for constructing another control (by the duality) such that there exist δ ∈ (0 , T ( M )) and v ∈ L ∞ (Ω × R + ), with � v � L ∞ ≤ M , such that Ω × (0 , T ∗ − δ ) , ∂ t y − ∆ y = χ ω v in ∂ Ω × (0 , T ∗ − δ ) , y = 0 on y ( x , 0) = y 0 ( x ) in Ω , y ( x , T ∗ − δ ) = 0 in Ω . This leads to a contradiction with the time optimality of T ( M ). V. J. Mizel, T. I. Seidman, An abstract bang-bang principle and time optimal boundary control of the heat equation, SICON, 1997. G. Wang, L ∞ -Null controllability for the heat equation and its consequences for the time optimal control problem, SICON, 2008. C. ZHANG Time optimal control problems 7 / 21
Motivations 1 Main results 2 Heat equations Abstract evolution equations An specific example Conclusion and further research 3 C. ZHANG Time optimal control problems 8 / 21
Space-time analyticity estimate Recall the following result in the last talk given by M. Santiago: Theorem Assume that △ 4 R ( q 0 ) (when non-empty) is real-analytic. Then, there are constants N and ρ , with 0 < ρ ≤ 1 , such that t e t ∆ f ( x ) | ≤ N ( t − s ) − n 4 e 8 R 2 / ( t − s ) | α | ! β ! x ∂ β | ∂ α � e s ∆ f � L 2 (Ω) , ( R ρ ) | α | (( t − s ) / 4) β when x ∈ B 2 R ( q 0 ) ∩ Ω , 0 ≤ s < t, α ∈ N n and β ≥ 0 . Here △ 4 R ( q 0 ) � B 4 R ( q 0 ) ∩ ∂ Ω. C. ZHANG Time optimal control problems 8 / 21
Propagation of estimate for real-analytic functions Theorem Assume that f : B 2 R ⊂ R n − → R is real-analytic in B 2 R verifying x f ( x ) | ≤ M | α | ! | ∂ α ( ρ R ) | α | , when x ∈ B 2 R , α ∈ N n , for some M > 0 and 0 < ρ ≤ 1 . Let E ⊂ B R be a measurable set with positive measure. Then, there are positive constants N = N ( ρ, | E | / | B R | ) and θ = θ ( ρ, | E | / | B R | ) , with θ ∈ (0 , 1) , such that � θ � � M 1 − θ . � f � L ∞ ( B R ) ≤ N — | f | dx E S. Vessella, A continuous dependence result in the analytic continuation problem, Forum Math., 11 (1999), 695–703. C. ZHANG Time optimal control problems 9 / 21
Main results Based on these two theorems and a telescoping series method introduced in the last talk, we have The observability estimates (1) and (2) over measurable sets of positive measure in both space and time variables are true. C. ZHANG Time optimal control problems 10 / 21
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