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PARAMETRIC OPTIMAL CONTROL PROBLEMS

UNIVERSIDADE de AVEIRO Departamento de Matematica, 2005. PARAMETRIC OPTIMAL CONTROL PROBLEMS. Olga Kostyukova. Institute of Mathematics National Academy of Sciences of Belarus Surganov Str.11, Minsk, 220072 e’mail kostyukova@im.bas-net.by. OUTLINE. Problem statement;

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PARAMETRIC OPTIMAL CONTROL PROBLEMS

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  1. UNIVERSIDADE de AVEIRO Departamento de Matematica, 2005 PARAMETRIC OPTIMAL CONTROL PROBLEMS Olga Kostyukova Institute of Mathematics National Academy of Sciences of Belarus Surganov Str.11, Minsk, 220072 e’mail kostyukova@im.bas-net.by

  2. OUTLINE • 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.

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

  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 problemOC(h),

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

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

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

  9. 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

  10. Suppose for a given we know • solution to problem • a vector • corresponding structure and defining elements The question is how to find is a sufficiently small right-side neighborhood of Here the point

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

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

  13. Optimal control for ОС(h):

  14. 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

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

  16. Old switchingfunction New switchingfunction

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

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

  19. B):How to determine

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

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

  22. Let (QP) have unique optimal plans We had problems: Solution of problemA):

  23. Solution of problemB):

  24. Theorem 3.Let h0be 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 is constructed by the rules optimal control

  25. 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.

  26. Results of these investigations are presented in the papers: • 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).

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