# Hélène Frankowska

Institut de Mathématiques de Jussieu – Paris Rive Gauche

CNRS and Université Sorbonne, Paris

**8 March 2018, 10:00, salle de séminaire du CERMICS**

**Value Function in deterministic optimal control: sensitivity relations of first and second order**

Value function is a useful tool of optimal control theory, since it provides sufficient optimality conditions and allows to construct optimal trajectories and optimal synthesis by considering an auxiliary problem on its epigraph. It is well known that, under mild assumptions, the value function $V(t,x)$ of the deterministic optimal control, when differentiable, is the only solution of an associated Hamilton-Jacobi (HJ) equation. It is also well known that it is not differentiable whenever for some initial conditions the optimal control problem has multiple minimizers. Then solutions of HJ are understood in the viscosity sense and, again, the value function is the unique solution. These generalized solutions are defined using notions of super/subdifferentials of the value function. Translated to the language of characteristics, non-smoothness of $V$ may be also seen as the presence of shocks.

If $\bar x$ is an optimal trajectory, then the first order necessary optimality conditions involve an adjoint variable $p(\cdot)$ that satisfies the maximum principle. It is well known that the following sensitivity relation holds true: $p(t)=-V_x(t,\bar x(t))$ and, if $V$ is not differentiable, then $-p(t)$ is an element of the superdifferential of $V(t,\cdot)$ at $\bar x(t)$. Similarly, $V_{xx}(t,\bar x(t))$ solve a matrix Riccati equation.

The aim of this talk is to discuss sensitivity relations involving second order jets (used to study Hamilton-Jacobi equations in stochastic optimal control while dealing with nonsmooth value functions). A geometric local sufficient condition preventing shocks will be also provided.

Some related results can be found in the following references:

- CANNARSA P. and FRANKOWSKA H. (2013)
*Local regularity of the value function in optimal control*, Systems and Control Letters, 62, 791–794 - BETTIOL P., FRANKOWSKA H. and VINTER R. (2015)
*Improved sensitivity relations in state constrained optimal control*, Applied Mathematics and Optimization, 71, 353–377 - CANNARSA P., FRANKOWSKA H. and SCARINCI T. (2015)
*Second-order sensitivity relations and regularity of the value function for Mayer’s problem in optimal control,*SIAM Journal on Control and Optimization, 53, 3642–3672 - FRANKOWSKA H. and HOEHENER D. (2017)
*Pointwise second order necessary optimality conditions**and second order sensitivity relations in optimal control*, J. of Diff. Eqs., 262, 5735-5772

# TBA

**22 March 2018, 10:00, salle de séminaire du CERMICS**

# Laura Grigori

Inria, LJLL

**5 April 2018, 10:00, salle de séminaire du CERMICS**

# Ioannis Stefanou

Navier

**19 April 2018, 10:00, salle de séminaire du CERMICS**

Archive of past seminars: here

Organizers: Antoine Levitt, Julien Reygner.