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Optimal control of nonlinear evolution equations

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In this paper, first we consider parametric control systems driven by nonlinear evolution equations defined on an evolution triple of spaces. The parametres are time-varying probability measures (Young measures) defined on a compact metric space. The appropriate optimization problem is a minimax control problem, in which the system analyst minimizes the maximum cost (risk). Under general hypotheses on the data we establish the existence of optimal controls.
Then we pass to nonparametric systems, which are governed by nonlinear evolution equations with nonmonotone operators. We prove two existence results for such evolution inclusions, which are of independent interest and extend significantly the results existing in the literature. Then we solve time-optimal and Meyer-type optimization problems. In Section 5, we derive necessary conditions for saddle point optimality in the minimax control problem. We conclude the paper with three examples of distributed parameter control systems.
  • National Technical University, Department of Mathematics, Zografou Campus, Athens 157 80, Greece
  • National Technical University, Department of Mathematics, Zografou Campus, Athens 157 80, Greece
  • [1] N.U. Ahmed and X. Xiang, Optimal control of infinite dimensional uncertain systems, J. Optim. Theory Appl 80 (1994), 261-272.
  • [2] S. Aizicovici and N.S. Papageorgiou, Infinite dimensional parametric optimal control systems, Japan Jour. Ind. Appl. Math. 10 (1993), 307-332.
  • [3] R. Ash, Real Analysis and Probability, Academic Press, New York 1972.
  • [4] H. Attouch and A. Damlamian, Problemes d'evolution dans les Hilberts et applications, J. Math. Pures Appl. 54 (1974), 53-74.
  • [5] E. Balder, Necessary and sufficient conditions for L₁ strong-weak lower semicontinuity of integral functionals, Nonlin. Anal-TMA 11 (1987), 1399-1404.
  • [6] H. Brezis, Operateurs Maximaux Monotones, North-Holland, Amsterdam 1973.
  • [7] F. Browder, Pseudomonotone operators and nonlinear elliptic boundary value problems on unbounded domains, Proc. Natl. Acad. Su USA 74 (1977), 2659-2661.
  • [8] L. Cesari, Existence of solutions and existence of optimal solutions, in: Mathematical Theories of Optimization, eds. J. Cecconi and T. Zolezzi, Springer, Berlin, Lecture Notes in Math. 979 (1983), 88-107.
  • [9] L. Cesari and S.H. Hou, Existence of solutions and existence of otpimal solutions, The quasilinear case, Rendiconti del Circolo Matematico di Palermo 34 (1985), 5-45.
  • [10] C. Dellacherie and P.A. Meyer, Probabilities and Potential, North Holland, Amsterdam 1978.
  • [11] J. Diestel and J. Uhl, Vector Measures, Math. Surveys 15, AMS, Providence, R.I (1977).
  • [12] J. Dugundji, Topology, Allyn and Bacon Inc, Boston 1966.
  • [13] I. Ekeland, On the variational principle, J. Math. Anal. 47 (1974), 324-353.
  • [14] A. Fryszkowski and J. Rzeżuchowski, Continuous version of Filippov-Ważewski relaxation theorem, J. Diff. Eqns 94 (1991), 254-265.
  • [15] J.-P. Gossez and V. Mustonen, Pseudomonotonicity and the Leray-Lions condition, Diff. and Integral Eqns 6 (1993), 37-45.
  • [16] N. Hirano, Nonlinear evolution equations with nonmonotone perturbations, Nonl. Anal - TMA 13 (1989), 599-609.
  • [17] S.H. Hou, Existence theorems of optimal control problems in Banach spaces, Nonlin. Anal - TMA 7 (1983), 239-257.
  • [18] S.H. Hou, Existence of solutions for a class of nonlinear control problems, J. Optim. Th. Apl. 66 (1990), 23-35.
  • [19] S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis Volume I, Theory, Kluwer, Dordrecht 1997.
  • [20] S. Hu and N.S. Papageorgiou, Time dependent subdifferential evolution inclusions and optimal control, Memoirs of the AMS (May 1998-in press).
  • [21] C. Ionescu-Tulcea, Topics in the Theory of Lifting, Springer Verlag, Berlin 1969.
  • [22] J.-L. Lions, Quelques Methodes de Resolution des Problemes aux Limites Non-Lineaires, Dunod, Paris 1969.
  • [23] J. Oxtoby, Measure and Category, Springer Verlag, New York 1971.
  • [24] N.S. Papageorgiou, Optimal control of nonlinear evolution equations, Public Math. Debrecen 41 (1992), 41-51.
  • [25] N.S. Papageorgiou, Optimization of parametric controlled nonlinear evolution equations, Yokohama Math. Jour 42 (1994), 107-120.
  • [26] N.S. Papageorgiou, Existence theory for nonlinear distibuted parameter optimal control problems, Japan Jour Indust. Appl. Math. 12 (1995), 457-485.
  • [27] N.S. Papageorgiou, Minimax control of nonlinear evolution equations, Comment. Math. Univ. Carolinae 36 (1995), 36-56.
  • [28] N.S. Papageorgiou, Boundary value problems and periodic solutions for semilinear evolution inclusions, Comment. Math. Univ. Carolinae 35 (1994), 325-336.
  • [29] N.S. Papageorgiou, On the existence of solutions for nonlinear parabolic problems with nonmonotone discontinuities, J. Math. Anal. Appl. 205 (1997), 434-453.
  • [30] N.S. Papageorgiou, A continuous version of the relaxation theorem for nonlinear evolution inclusions, Kodai Math. Jour. 18 (1995), 169-186.
  • [31] N.S. Papageorgiou, F. Papalini and F. Renzacci, Existence of solutions and periodic solutions for nonlinear evolution inclusions, Rendiconti del Circolo Matematico di Palermo, to appear.
  • [32] N.S. Papageorgiou and N. Shahzad, Existence and strong relaxation theorems for nonlinear evolution inclusions, Yokohama Math. Jour. 43 (1995), 73-87.
  • [33] N.S. Papageorgiou and N. Shahzad, Properties of the solution set of nonlinear evolution inclusions, Appl. Math. Otpim. 36 (1997), 1-20.
  • [34] K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, New York 1967.
  • [35] J. Simon, Compact sets in the space $L^{p}(0,T;B)$, Ann Mat. Pura ed Appl. 146 (1987), 65-96.
  • [36] B.A. Ton, Nonlinear evolution equations in Banach spaces, J. Diff. Eqns 9 (1971), 608-618.
  • [37] E. Zeidler, Nonlinear Functional Analysis and Applications II, Springer Verlag, New York 1990.
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