The main goal of this project is to start an active collaboration between former Soviet Union and european researchers in the field of control theory and differential games creating a network. The theoretical point of view and the development of numerical schemes to solve real problems arising in applications will be both emphasized.

ORGANIZATION

TIME SCHEDULE

BUDGET

SCIENTIFIC PROGRAM

1. Geometric methods in optimal control
The main goal is to investigate the characterization of extremal and optimal arcs in control problems.

• obtaining second order necessary and sufficient conditions for weak local optimality in presence of state-control constraints.
• extending the classical theory of fields of extremals to constrained control problems in order to obtain sufficient optimality conditions for strong local minima.
• to develop an Hamilton-Jacobi theory for problem with variable endpoints.
• investigate high-order variations of control systems near a reference trajectory. There are invariant expressions of the variations via iterated Lie brackets of vector fields which form the systems. The aim of the research is to recognize Lie bracket relations, responsible for the most important local properties of the system, in particular, a local optimality of the reference trajectory.
• to use methods of symplectic geometry to design optimal synthesis.

2. Necessary and sufficient conditions for optimal solutions
The theory of necessary and sufficient conditions for optimal solutions of problems of optimal control and differential games will be investigated looking in particular for conditions which can be successfully applied in the develpment of numerical methods based on the Pontryagin principle.

3. Sufficient conditions for nonlocal controllability
We will study sufficient conditions for nonlocal controllability of nonlinear ordinary differential equations (ODEs). To obtaining these conditions we introduce simpler ODEs, which satisfies Wazhevski like conditions, and some mapping from the state space of these auxiliary ODEs. When some additional conditions are satisfied, this mapping is called vector Lyapunov function (VLF). Under these conditions controllability of auxillary ODE implies the same property for the original ODE. A constructive procedure to find VLF will be investigated. Similary, we will consider controllability under phase restrictions and/or persistent perturbations and other properties of optimality.

4. Nonlinear optimal control problems with impulsive and singular controls.
We intend to investigate control problems with unbounded strategies from various points of view: definitions of space-trajectories, existence problems, dynamic progamming, synthesis, possible relations with singular perturbations.
One approach is based on the method of the discontinuos time change that gives the possibility to obtain the representation for optimal paths by the nonlinear differential equations with a measure and to derive the optimality conditions in the maximum principle form. We plan to extend this method to nonlinear discrete-continuous dynamical systems with general types of constraints and performance criterions. Another approach is connected with the construction of a link between impulsive controls and the corresponding singularly perturbed optimal control problems on the base of sufficient optimality conditions of absolute minimum. We plan to study the evolution laws for attainable sets of controlled differential equations by a characterization in terms of generalized derivatives of a function of bounded variation.

5. Optimal control for systems with delay and incomplete state information.
It is proposed to investigate approximate and numerical methods for the following optimal control problems for Functional Differential Equations (FDEs in short):

• Bilinear systems with delay controls,
• Control of discontinuos FDEs
• Adaptive control of FDEs
• Control of FDEs with jump coefficients
• Control and stabilisation of stocastic FDEs
• Applications in ecology , medicine and flight mechanics
• Optimal flitering ,interpolation and smoothing of FDEs in delays in observation.

6. Application of multivalued analysis methods and variational optimality conditions to control problems
The research will focus on the following problems:

• determination of the variational maximum principle (the first order optimality condition) for optimal impulsive control problems with phase restrictions and its applications to dynamical models in economics.
• quadratic necessary and sufficient optimality conditions for impulsive-singular processes with phase restrictions
• investigation of the methods of linear-quadratic approximations in optimal control problems
• proving the existence of optimal solutions in problems discribed by various kind of differential inclusions without convexity assumptions
• Wellposedness of optimal control problems.

7. Viscosity solutions of Hamilton-Jacobi equations.
The research will focus on further developments in the theory of weak solutions (in the viscosity sense) of Hamilton-Jacobi equation and, in particular, on the theory of discontinuous viscosity solutions.

8. Viability theory in control and differential games .
We plan to study a characterization of viability kernels for constrained control problems and of the discriminating victory domains for differential games. This study will be also applied to the construction of algorithms for the value function of control problems with state constraints and for the approximation of victory domains.

9. Zero-sum differential games .
The goal of this research is to compare and to unify, if possible, the approach of the FSU school to zero-sum differential games (Krassovski-Subbotin) with the approaches developed in Western countries. In particular the problem for games with barriers, where the value function is discontinuous, is quite interesting and is related to some open problems in the theory of viscosity and minimax solutions of H-J equations. To this end, one line of the research is the development of the approach according to which the generalized solution (called the minimax one) is assumed to be weakly invariant with respect to "characteristic" differential inclusions, i.e. for any fixed point of the graph of minimax solution there exist viable trajectories of these differential inclusions that go within the graph through this point. This definition can be considered as a reduction or relaxation of the classical method of characteristics.\par Criteria for weak invariance of minimax solutions will be considered. These criteria can be formulated in terms of various constructions of nonsmooth and convex analysis: directional derivatives, contingent cones, sub and super-differentials, etc. Studying these criteria promotes better understanding of the essence of the duality inherent to the Hamilton-Jacobi equations. This duality implies, in particular, the equivalence of minimax and viscosity solutions. Methods of the theory of minimax and viscosity solutions will be applied in the theory of control and differential games for solving problems of feedback control.

10. Dynamic games with partial information.
No general theory still exists for games with non-conventional information structure (incomplete, disymetric or delayed information etc...). We shall investigate several particular games in order to gain insights on these problems. On another side we establish the equivalence principle for nonlinear partial information min-max control problems.

11. Theory and approximation of optimal control problems for distributed systems
We intend to study the characterization of the value function related to optimal control problems for partial differential equations (mainly parabolic equations). We will consider both the problems with distributed control and boundary controls. Numerical methods based on the open-loop and the closed-loop approach will be developped.

12. Numerical solution of Hamilton-Jacobi and Isaacs equations
We plan to develop algorithms for the approximate solution of optimal control problems and differential games by the dynamic programming approach. We will use the theory of viscosity solutions to establish the convergence and the rate of convergence of the algorithms for the value function. Further investigations will concern the convergence of optimal trajectories and of feedback controls, the development of high-order schemes and domain decomposition strategies. This program will be developped for impulsive and singular control problems, problems with state constraints, pursuit-evasion games and stochastic control problems. Since several useful discrete schemes related to the the approximation of dynamic games have an interpretation in terms of stochastic games with a countable state space, we further investigate the problem of finite state approximations of these games.

13. Approximation methods for optimal control problems
The interest in Hamilton-Jacobi Theory has renewed in the last years by the development of modern high speed computing techniques. This makes possible, to solve a variety of optimal control problems for a wide range of applications. Also stabilization of nonlinear control systems can profit from the theory and numerics of Hamilton-Jacobi equations. Difficult problems of only partially stabilizable systems occur e.g. in the control of turbulence as shown recently by Holmes et al. Such problems are not solvable by classical control methods. Instead Hamilton-Jacobi theory and Lyapunov exponents yield a possible alternative in order to compute numerically stabilizing feedbacks. All aspects of this problem, spectral theory of time varying matrices, theory and numerics of Hamilton-Jacobi equations, feedback stabilization and geometric control theory with its connections to the theory of dynamical system will be addressed.
A different approach to the solution of optimal control problems is the so called improving method which contain an iterative process for the successive minimization of a functional. We intend to develop the improving methods based on the extension principle and V.F.Krotov's sufficient optimality conditions. These methods allow to solve the control problems for which well-established methods are not suitable. New methods for some classes of control problems (degenerated, distributed parameter etc.) using nonlinear transformations and reachable sets estimates will also be developped. Theoretical investigations of the algorithms properties, in particular on relaxation and convergence properties, and computational experiments will be part of the project. Finally, numerical methods based on necessary and sufficient conditions for optimal solutions of problems of optimal control and differential games will also be investigated. We plan to compare these methods with the schemes based on dynamic programming and eventually try to couple the two approaches.

14. Applications to economics, robotics, motion planning problems and aero-space, mechanical and chemical engineering.
The above mentioned results and the algorithms will be applied to the numerical solution of model problems arising in real applications:

• a) Applications to economics: optimal growth theory, economic stabilization, advertising models, portfolio models.
• b) Motion planning problem. Given two oriented points in the plane, we consider the problem of finding the shortest path joining them under some constraint. In the simplest case where the curvature of the path has to remain bounded, this problem has been solved recently by Dubins in the no-cusp case, and by Reeds and Shepp otherwise. It has been shown in that the minimum principle of Pontryagin, together with some geometrical arguments, leads to the full characterization of the solutions. Moreover, a complete synthesis has been obtained. If the derivative of the curvature is to be kept bounded, we still don't know if there exists a "regular" shortest path for the problem with no-cusp. However, suboptimal trajectories have been built. Such problems are of great interest in robotics for the motion planning of car-like robots, and especially their generalization to the 3-dimensional case or to a situation with obstacles.
• c) Optimal re-entry of a space shuttle and problems of optimal guidance.