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.

1 day in Western Countries = 83 ECU

1 day in Moscow = 33 ECU

1 day in any other place in NIS = 25 ECU

The budget will be administrated by the country coordinators and by the project coordinator according to the the rules of network (see the provisional Network Agreement Encl. 2).

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

We are interested in the following 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.