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June 2024, 2016 Institut Henri Poincar e, Paris Nonlinear Partial Differential Equations and Applications, in Honor of Professor Jean-Michel Corons 60th Birthday t f a An Invitation to Control Theory of Stochastic

  1. June 20–24, 2016 · · · Institut Henri Poincar´ e, Paris “Nonlinear Partial Differential Equations and Applications, in Honor of Professor Jean-Michel Coron’s 60th Birthday” t f a An Invitation to Control Theory of Stochastic Distributed Parameter Systems r D Xu Zhang School of Mathematics Sichuan University, China zhang xu@scu.edu.cn • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

  2. t f Outline: a 1. Introduction r D 2. Controllability for stochastic DPS 3. Optimal control for stochastic DPS • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

  3. 1. Introduction t ♦ What is stochastic DPS and its control? f Systems governed by stochastic differential equations in infinite dimensions. a • Stochastic differential equations with delays; r • Stochastic PDEs; D • Random PDEs, i.e., PDEs with random parameters; • · · · · · · Control: One hopes to change the system’s dynamics, by means of a suitable way. Two fundamental issues: Controllability and Optimal Control. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

  4. ♦ Why should study control theory for stochastic DP- t S? f Claim: For control theory, it is now stochastic DPS’ a turn! Why? • Control Theory for ODE systems: Relatively ma- r ture, many classics. D L.S. Pontryagin: Maximum Principle; R. Bellman: Dynamic Programming and HJB Equa- tions; R.E. Kalman: LQ Problem and Filter Theory. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

  5. • Control Theory for DPS: Many results (many many papers, many books), still quite active. t Pioneers: Yu V. Egorov, H. O. Fattorini, J.-L. Lions, D. L. Russell...... f Early books: a [1] J.-L. Lions. Optimal Control of Systems Gov- erned by Partial Differential Equations. Springer- r Verlag, 1971. D [2] R.F. Curtain and A.J. Pritchard. Infinite Dimen- sional Linear Systems Theory. Springer-Verlag, 1978. “Recent” books: [1] X. Li and J. Yong. Optimal Control Theory for Infinite-Dimensional Systems. Birkh¨ auser, 1995. [2] J.-M. Coron. Control and Nonlinearity. American Mathematical Society, 2007. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

  6. • Control theory for stochastic systems in finite dimensions (i.e., Stochastic ODEs): Many works, t closely related to mathematical finance. f Important works: A. Bensoussan, J.-M. Bismut, W. a H. Fleming, H.J. Kushner, S. Peng...... Classical books: r [1] W. H. Fleming and H. M. Soner. Controlled D Markov Processes and Viscosity Solutions. Springer- Verlag, 1992. [2] J. Yong and X. Zhou. Stochastic Controls: Hamil- tonian Systems and HJB Equations. Springer-Verlag, 1999. Controllability theory is NOT well-developed. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

  7. • Control theory for stochastic DPS: Almost at its very beginning stage. Still an ugly duckling! t Very few works. f Only three books (The first two addressed mainly to a some slightly different topics): [1] A. Bashirov. Partially Observable Linear Systems r Under Dependent Noises. Birkh¨ auser Verlag, 2003. D [2] P. S. Knopov and O. N. Deriyeva. Estimation and Control Problems for Stochastic Partial Differential Equations. Springer, 2013. [3] Q. L¨ u and X. Zhang. General Pontryagin- Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimen- sions. Springer, 2014. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

  8. The most general control system in the framework of t classical physics. f The study of this field may provide some useful hints for that of quantum control systems. a It will eventually become a white swan! r ♦ Why the study of control theory for stochastic DPS D is difficult? • Very few are known for stochastic PDEs. • Both the formulation of stochastic control problems and the tools to solve them may differ considerably from their deterministic counterpart. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

  9. • One will meet substantially difficulties in the study t of control problems for stochastic DPS. f Unlike the deterministic setting, the solution to an a SDE/SPDE is usually non-differentiable with respect to the variable with noise. r The usual compactness embedding result fails to be true for the solution spaces related to SDEs/SPDEs. D The “time 0 in the stochastic setting is not reversible, even for stochastic hyperbolic equations. Generally, stochastic control problems cannot be re- duced to deterministic ones. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

  10. 2. Controllability for stochastic DPS t ♦ Controllability for stochastic ODEs f • The deterministic setting a Consider the following controlled (ODE) system:  r d  t ∈ [0 , T ] , dty = Ay + Bu, D (1)  y (0) = y 0 , where A ∈ R n × n , B ∈ R n × m , T > 0 . System (1) is said to be controllable on (0 , T ) if for any y 0 , y 1 ∈ R n , there exists a u ∈ L 1 (0 , T ; R m ) such that y ( T ) = y 1 . • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

  11. Put � T e At BB ∗ e A ∗ t dt. G T = t 0 f Theorem µ If the system (1) is controllable on (0 , T ) , a then det G T � = 0 . Moreover, for any y 0 , y 1 ∈ R n , the control r u ∗ ( t ) = − B ∗ e A ∗ ( T − t ) G − 1 T ( e AT y 0 − y 1 ) D transfers y 0 to y 1 at time T . Clearly, if (1) is controllable on (0 , T ) (by means of L 1 -(in time) controls), then the same controllability can be achieved by using analytic-(in time) controls. We shall see a completely different phenomenon in the simplest stochastic situation. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

  12. • The stochastic setting (Ω , F , {F t } t ≥ 0 , P ) : a complete filtered probability t space on which a one dimensional standard Brownian f motion { B ( t ) } t ≥ 0 is defined. a H : a Banach space, and write F = {F t } t ≥ 0 . L 2 F (0 , T ; H ) : the Banach space consisting of r all H -valued F -adapted processes X ( · ) such that D E ( | X ( · ) | 2 L 2 (0 ,T ; H ) ) < ∞ , with the canonical norm; Similaryly, L ∞ F (0 , T ; H ) , L 2 F (Ω; C ([0 , T ]; H )) , etc. The filtration F plays a crucial role, and it represents the “information” that one has at each time t . For SDE (in the Itˆ o sense), one needs to use adapted processes X ( · ) , i.e., ∀ t , the r.v. X ( t ) is F t -measurable. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

  13. Consider a one-dimensional controlled stochastic dif- t ferential equation: f dx ( t ) = [ bx ( t ) + u ( t )] dt + σdB ( t ) , (2) a with b and σ being given constants. We say that the system (2) is exactly controllable if for any x 0 ∈ R r and x T ∈ L 2 F T (Ω; R ) , there exists a control u ( · ) ∈ L 1 F (0 , T ; L 2 (Ω; R )) such that the corresponding solu- D tion x ( · ) satisfies x (0) = x 0 and x ( T ) = x T . Q. L¨ u, J. Yong and X. Zhang (JEMS, 2012) showed that the system (2) is exactly controllable at any time T > 0 (by means of L 1 F (0 , T ; L 2 (Ω; R )) -controls). • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

  14. t On the other hand, surprisingly, in virtue of a result f by S. Peng (Progr. Natur. Sci., 1994), the system (2) is NOT exactly controllable if one restricts to use a admissible controls u ( · ) in L 2 F (0 , T ; L 2 (Ω; R )) ! r Q. L¨ u, J. Yong and X. Zhang (JEMS, 2012) showed that the system (2) is NOT exactly controllable, ei- D ther provided that one uses admissible controls u ( · ) in L p F (0 , T ; L 2 (Ω; R )) for any p ∈ (1 , ∞ ] . This leads to a corrected formulation for the exact controllability of stochastic differential equations, as presented below. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

  15. • Definition of exact controllability Consider a linear stochastic differential equation: t � � � � � f dy = Ay + Bu dt + Cy + Du dB ( t ) , t ∈ [0 , T ] , a y (0) = y 0 ∈ R n , (3) r where A, C ∈ R n × n and B, D ∈ R n × m . No univer- sally accepted notion for stochastic controllability! D Definition : System (3) is said to be exactly control- lable if for any y 0 ∈ R n and y T ∈ L 2 F T (Ω; R n ) , ∃ a con- trol u ( · ) ∈ L 1 F (0 , T ; L 2 (Ω; R m )) such that Du ( · , ω ) ∈ L 2 (0 , T ; R n ) , a.e. ω ∈ Ω and the corresponding solu- tion y ( · ) to (3) satisfies y ( T ) = y T . • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

  16. t Though the above definition seems to be a reasonable notion for exact controllability of stochastic differen- f tial equations, a complete study on this problem is a still under consideration and it does not seem to be easy. r When n > 1 , the controllability for the linear system D (5) is in general unclear. Compared to the deterministic case, the controllabil- ity/ observability for stochastic differential equations is at its / enfant 0 stage. • First • Prev • Next • Last • Go Back • Full Screen • Close • Quit

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