2. Problem statement; Solution structure and defining elements; Solution properties in a neighborhood of regular point; Solution properties in a neighborhood of irregular point: construction of new Lagrange vector; construction of new structure and defining elements; Generalizations. OUTLINE

3. Family of parametric optimal control problems: are given functions, is a parameter. Problem statement

4. Optimal control and trajectory for problem The aims of the talk are to investigate dependence of the performance index and on the parameter h; to describe rules for constructing solutions to

5. Terminal control problem OC(h) is solution to the problem OC(h),

6. Maximum Principle to be optimal in ОС (h)In order for admissible control it is necessary and sufficient that a vector exists such that the following conditions are fulfilled Here is a solution to system

7. Denote bythe set of all vectors y, satisfying (2), (3) The setis not empty and is bounded for and consider the mapping The mapping (4) is upper semi-continuous. Let Denote by the corresponding switching function.

8. Zeroes of the switching function: Active index sets: Double zeroes:

10. Solution structure: Defining elements: Regularity conditions for solution(for parameter h) Lemma 1. Property of regularity (or irregularity) for control does not depend on a choice of a vector

11. Suppose for a givenwe know solutionto problem a vector corresponding structureand defining elements The question is how to find is a sufficiently small right-side neighborhood of the point Here

12. Solution Properties in a Neighborhood of Regular Point Solution structure does not change:

13. Defining elements with initial conditions are uniquely found from defining equations where

14. Optimal control for ОС(h):

15. Construction of solutions in neighborhood of irregular point The set consists of more than one vector. The first Problem: How to find The second Problem: How to find

16. Theorem 1. The vector is a solution to the problem The problem (SI) is linear semi-infinite programming problem. The setis not empty and is bounded the problem (SI) has a solution. Suppose that the problem (SI) has a unique solution

17. New switching function Old switching function

18. A) What indicesare in the new set of active B) How many switching points will new optimal control indices have?

19. Form the index sets It is true that ? A): How to determine

20. B): How to determine

23. Using known vector and sets form quadratic programming problem (QP):

24. Theorem 2. Suppose that there exist finite derivatives Then the problem (QP) has a solution which can be uniquely found using derivatives Then derivatives are uniquely calculated by Suppose the problem (QP) has a unique optimal solution: primaland dual

25. Let (QP) have unique optimal plans We had problems: Solution of problem A):

26. Solution of problem B):

28. Theorem 3. Let h 0 be an irregular point and the problem (QP) have a unique solution. problems ОС(h) have regular solutions with constant structure defining elements Q(h) are uniquely found from optimal control is constructed by the rules

29. On the base of these results the following problems are investigated and solved differentiability of performance index and solutions to problems path-following (continuation) methods for constructing solutions to a family of optimal control problems; fast algorithms for corrections of solutions to perturbed problems construction of feedback control.

30. Kostyukova O.I. Properties of solutions to a parametric linear-quadratic optimal control problem in neighborhood of an irregular point. // Comp. Math. and Math. Physics, Vol. 43, No 9, 1310-1319 (2003). Kostyukova O.I. Parametric optimal control problems with a variable index. Comp. Math. and Math. Physics, Vol. 43, No 1, 24-39 (2003). Kostyukova, Olga; Kostina, Ekaterina. Analysis of properties of the solutions to parametric time-optimal problems. // Comput. Optim. Appl. 26, No.3, 285-326 (2003). Kostyukova, O.I. A parametric convex optimal control problem for a linear system. // J. Appl. Math. Mech. 66, No.2, 187-199 (2002). Kostyukova, O.I. An algorithm for solving optimal control problems. // Comput. Math. and Math. Phys. 39, No.4, 545-559 (1999). Kostyukova, O.I. Investigation of solutions of a family of linear optimal control problems depending on a parameter. // Differ. Equations 34, No.2, 200-207 (1998). Results of these investigations are presented in the papers: