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Optimal control and differential games: theory, numerical methods and applications

During the first two years the Network has supported several short term visits devoted to establish contacts (in both directions) within the teams of the project.
The main scientific results obtained by the researchers of the network are:

-Second order necessary and sufficient conditions for weak local optimality in presence of state-control constraints (Firenze, Steklov Inst.)
-Wellposedness of optimal control problems. (Genova)
-Variable structure control methods and applications to uncertain control systems.(Genova)
-Stable behaviour of NaBh equilibria of non-co-operative games in strategic form. Grid schemes to design a dynamic Nash equilibrium in evolutionary games.(Ekaterinburg, Genova)
-Characterisation of the value functions of different control problems ( e.g. optimal time with and without controllability and coercivity assumptions, H-infinity control problems, differential games) as unique solutions of the corresponding boundary value problems. Hamilton-Jacobi theory for finite and infinite dimensional control problems. (Padova, INRIA- Sophia Antipolis, MIEM Moscow, Roma "La Sapienza", Roma "Tor Vergata").
-Control problems with unbounded controls and possibly without coercivity assumptions. (Padova, Roma "La Sapienza", IPPI-RAS)
-Calculus of Variations: one-dimensional and multidimensional problems are investigated, especially in the case without convexity assumptions. (Irkutsk, Padova)
-Stochastic control and degenerate second order problems. (Roma "La Sapienza", IPPI Moscow)
-Numerical methods for Hamilton-Jacobi and Isaac's equations, high-order approximation schemes. (Ekaterinburg, Irkutsk, Roma "La Sapienza", Roma "Tor Vergata")
-Numerical methods for the control of partial differential equations and applications. (Roma "La Sapienza", Roma "Tor Vergata")
-Optimal design and shape-optimisation. (Roma "La Sapienza")
-Set-valued analysis and Control Theory of Viability Theory. Analysis of Dynamical Systems with States Constraints, of Control of systems with uncertainty (Robustness) and of Differential Games. Applications to Mathematical Economics and Demography. (Irkutsk, Paris IX Dauphine)
-Improved solution techniques for optimal control problems and differential games by means of an indirect multiple shooting method (Clausthal, Munich) Improved implementations of a direct collocation method (Munich)
-Two new implementations of a direct multiple shooting method for parameter identification problems in differential algebraic systems (Munich) and for large scale systems of differential algebraic equations (Clausthal)
-A new method for the synthesis of closed loop controls for optimal control and differential game problems via local Taylor expansions and neural networks (Clausthal)
-A new method for the real-time computation of closed loop controls for optimal control problems via linearisation around reference trajectories and by means of parallel computing (Munich)
-Real-life applications in aerospace engineering, robotics, chemical and mechanical engineering, telecommunication, economy (Clausthal, Munich)
-Theoretical investigations on sensitivity and feedback controls (Clausthal)
-Asymptotic expansions of singularly perturbed optimal control problems. (Pereslavl- Zalessky)
-Applications of the computer algebra system REDUCE in control theory and construction of asymptotics of attainability sets estimates for linear singular perturbed optimal control problems. (Irkutsk, Pereslavl- Zalessky)
-Games Theory with several pursuers and one evader. (Moscow, INRIA-Sophia Antipolis)
-Investigation of optimisation problems for discrete-continuous deterministic and stochastic systems, described by differential equations with measure coefficients. (IPPI, Moscow)
-Estimation of the state for discrete-continuous systems. Explicit solution of a continuous time filtering problem, where the coefficients in the dynamical equations are driven by a finite-state Markov process. (IPPI, Moscow)
-Robustness of the discrete-continuous systems with respect to the approximation of impulsive input. (Pereslavl-Zalessky)
-Characterisation of optimal open-loop and closed-loop (feedback) controls for the optimal control of non-linear dynamical systems with control and state constraints. Geometric control theory and methods of perturbations of extremal problems with constraints. (Firenze, Steklov Institute of Mathematics, Moscow)
-Numerical methods for constructing attainability sets of non-linear control systems and for finding level sets of the value function (Augsburg, Ekaterinburg, Roma "La Sapienza")
-Properties of stability (viability) in differential games. (Paris IX Dauphine, Ekaterinburg)
-Non-linear dynamic models of interaction for ecological communities, chemostat models and models of laboratory's test tubes, described by non-linear equations with delay.(MIEM, Moscow)
-Liapunov's functionals for systems with delay. (MIEM, Moscow)

Finally, it should be mentioned that several dissertations at PhD level were prepared under the scientific supervision of participants to the network. The research activity within the Network has generated some fruitful and new collaborations which will produce in the future more focused research projects.

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Università di Roma "La Sapienza"
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