Preface ......................................................... 7
Chapter 1. Some generalizations of the Dubovitskii-Milyutin
method ............................................... 9
1.1 Introductory remarks ....................................... 9
1.2 Definitions of cones. Separation theorem .................. 11
1.3 Statement of Pareto optimal problems ...................... 14
1.4 Necessary conditions for local Pareto optimum.
Generalized Dubovitskii-Milyutin theorem .................. 15
1.5 Necessary and sufficient conditions for global Pareto
optimum ................................................... 19
1.6 Problem "scalarization" ................................... 22
1.7 The Salukwadze optimum .................................... 24
1.8 Case s = 1 ................................................ 26
Chapter 2. Optimal control of parabolic systems ................ 28
2.1 Pareto optimal control problem for a cascade of
parabolic systems ......................................... 28
2.2 Quadratic Pareto optimal control for a parabolic
equation .................................................. 35
2.3 Optimal control problems with non-standard functionals
and time delay ............................................ 41
2.3.1 Parabolic equation with time delay ................. 42
2.3.2 Statement of optimal control problem. Optimality
conditions ......................................... 43
Chapter 3. Optimal control of systems with infinite number
of variables ........................................ 50
3.1 Introductory remarks ...................................... 50
3.1.1 Some functional spaces ............................. 50
3.1.2 Petrowsky type equation with an infinite number
of variables ....................................... 52
3.2 Optimal control problem for Petrowsky type equation.
Optimality conditions ..................................... 53
3.3 Time-optimal control problem for parabolic equations ...... 59
3.3.1 An equivalent optimization problem ................. 59
3.3.2 Optimality conditions .............................. 60
Chapter 4. Numerical example ................................... 65
4.1 Problem statement ......................................... 65
4.2 Projective gradient method ................................ 67
4.3 Application of projective gradient method to optimal
control problem ........................................... 67
4.4 Approximation of optimal control problem .................. 68
4.5 Numerical results ......................................... 70
Final remarks .................................................. 72
Appendix A. Programs in MAPLE V ................................ 73
A.l Program PARETOl.MS ................................... 74
A.2 Program PARETO2.MS ................................... 78
References ..................................................... 84
Index .......................................................... 89
Streszczenie ................................................... 90
Резюме ......................................................... 92
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