**09. October 2018** (CMAP, Ecole polytechnique, Salle de conférence)

**10h00:****A higher-order Maximum Principle for optimal unbouded control problems**

Maria Soledad Aronna (Escola de Matemática Aplicada, FGV, Rio de Janeiro, Brazil)

**Abstract**: We consider control systems governed by nonlinear O.D.E.’s in which the derivative of the control is involved. Such equations appear e.g. in Classical Mechanics, where some components of the systems act as a control and the other as dependent state variables. The control is this framework is usually called “impulsive” or “unbounded”, and it is allowed to be a (discontinuous) bounded variation function, whose derivative gives the system an impulsive character. For this class of equations, we adopt the concept of “graph completion solution”, that was introduced by A. Bressan and F. Rampazzo in the 90’s. We consider an optimal control problem in the Mayer form, with general control and final state constraints, for which we prove a maximum principle and higher-order necessary conditions in terms of the adjoint state and the Lie brackets of the involved vector fields.

Joint work with Monica Motta and Franco Rampazzo (Università di Padova, Italy).

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**11h00: A Second-order analysis for optimal control problems with partial differential equations**

Axel Kröner (Humboldt Universität, Berlin, Germany)

**Abstract**: In this talk we consider second order optimality conditions for bilinear optimal control problems governed by strongly continuous semigroups with the control entering linearly in the cost function. We derive first and second order optimality conditions, taking advantage of the Goh transformation. We then apply the results to heat, (damped) wave, and Schr\”odinger equations and discuss extensions to optimal control of the semilinear heat equation with additional constraints on the state.

Joint work with Franz Bethke (Humboldt Universität, Berlin, Germany), Frédéric Bonnans (INRIA) and Maria Soledad Aronna (Escola de Matemática Aplicada, FGV, Rio de Janeiro, Brazil).

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**25. September 2018**, 11h (CMAP, Ecole polytechnique, Salle de conférence)

**Taylor expansions of the value function associated with stabilization problems
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Laurent Pfeiffer (Universität Graz, Austria)

**Abstract**: It has been known for a long time that the value function associated with stabilization problems can be expanded in a Taylor series, where the involved multilinear forms are characterized by specific equations. Solving these equations allows to approximate locally the value function, without solving the Hamilton-Jacobi-Bellman equation. Taylor approximations can be used first to design polynomial feedback laws. They can also be used to improve the receding-horizon algorithm. In this talk, I will present recent theoretical results dealing with these two numerical approaches and give numerical results for a control problem of the Fokker-Planck equation.