×

Optimal feedback control for a semilinear evolution equation. (English) Zbl 0817.49002

The authors study the semilinear control system \[ \begin{aligned} y_ j'(t) & = A_ j(t) y_ j(t)+ f_ j(t, y_ 1(t),\dots, y_ n(t),\;u(t))\qquad (1\leq j\leq n)\\ u(t) &\in V(t, y_ 1(t),\dots, y_ n(t)),\end{aligned} \] where \(y_ j(t)\in\) separable Banach space \(X_ j\) and \(u(t)\in\) Banach space \(Z\); \(V\) is a feedback set-valued map. The object is to construct a feedback solution of the problem of minimizing a cost functional subject to constraints on the initial and final point of the trajectory.

MSC:

49J15 Existence theories for optimal control problems involving ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
47H04 Set-valued operators
47J05 Equations involving nonlinear operators (general)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Nistri, P., Obukhovskii, V. V., andZecca, P.,On the Solvability of Systems of Inclusions Involving Noncompact Operators, Transactions of the American Mathematical Society, Vol. 342, pp. 543–562, 1994. · Zbl 0793.47050 · doi:10.2307/2154640
[2] Conti, G., Nistri, P., andZecca, P.,Systems of Set-Valued Equations in Banach Spaces, Delay Differential Equations and Dynamical Systems, Edited by S. Busemberg and M. Martelli, Springer Verlag, Berlin, Germany, pp. 98–109, 1991. · Zbl 0831.34068
[3] Conti, G., Nistri, P., andZecca, P.,Systems of Multivalued Equations for Solving Controllability Problems, Pure Mathematics and Applications, Series B, Vol. 2, pp. 215–225, 1991. · Zbl 0772.49007
[4] Akhmerov, R. R., Kamenskii, M. I., Potapov, A. S., Rodkina, A. E., andSadovskii, B. N.,Measure of Noncompactness and Condensing Operators, Birkhauser Verlag, Berlin, Garmany, 1992. · Zbl 0748.47045
[5] Borisovich, Yu. G., Gel’man, B. D., Myshkis, A. D., andObukhovskii, V. V.,Topological Methods in the Fixed-Point Theory of Multivalued Maps, Russian Mathematical Surveys, Vol. 35, pp. 65–143, 1980. · Zbl 0464.55003 · doi:10.1070/RM1980v035n01ABEH001548
[6] Borisovich, Yu. G., Gel’man, B. D., Myshkis, A. D., andObukhovskii, V. V.,Multivalued Mappings, Journal of Soviet Mathematics, Vol. 24, pp. 719–791, 1984. · Zbl 0529.54013 · doi:10.1007/BF01305758
[7] Obukhovskii, V. V.,Semilinear Functional-Differential Inclusions in a Banach Space and Controlled Parabolic Systems, Soviet Journal Automat. Inform. Sci., Vol. 24, pp. 71–79, 1992.
[8] Górniewicz, L.,Homological Methods in Fixed-Point Theory of Multivalued Mappings, Dissértations Mathématiques, Vol. 129, pp. 1–71, 1976.
[9] Obukhovskii, V. V.,On the Topological Degree for a Class of Noncompact Multivalued Mappings, Funktsional’nyi Analiz (Ul’yanovsk), Vol. 23, pp. 82–93, 1984 (in Russian).
[10] Borisovich, Yu. G., Gel’man, B. D., Myshkis, A. D., andObukhovskii, V. V.,Multivalued Analysis and Operator Inclusions, Journal of Soviet Mathematics, Vol. 39, pp. 2772–2811, 1987. · Zbl 0725.54015 · doi:10.1007/BF01127054
[11] De Blasi, F.,Characteristics of Certain Classes of Semicontinuous Multifunctions by Continuous Approximations, Journal of Mathematical Analysis and Applications, Vol. 106, pp. 1–18, 1985. · Zbl 0574.54012 · doi:10.1016/0022-247X(85)90126-X
[12] Kamenskii, M. I., andObukhovskii, V. V.,On the Translation Operator along Trajectories of Solutions of Semilinear Differential Inclusions and Control Systems in Banach Spaces, Proceedings of the Conference on Qualitative Methods in Boundary-Value Problems, Voronezh, Russia, p. 68, 1992.
[13] Papageorgiou, N. S.,On Multivalued Evolution Equations and Differential Inclusions in Banach Spaces, Commentarii Mathematici Universitatis Sancti Pauli, Vol. 36, pp. 21–39, 1987. · Zbl 0641.47052
[14] Bryszewski, J., andGorniewicz, L.,A Poincaré-Type Coincidence Theorem for Multivalued Maps, Bullettin de l’Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, Vol. 24, pp. 593–598, 1976. · Zbl 0332.54016
[15] Kamenskii, M. I., andObukhovskii, V. V.,On Periodic Solutions of Differential Inclusions with Unbounded Operators in Banach Spaces, Zbornik Radova Prirodno-Matematichko Fakulteta, Serija za Matematiku, Vol. 21, pp. 173–191, 1991.
[16] Kamenskii, M. I., andObukhovskii, V. V.,Condensing Multioperators and Periodic Solutions of Parabolic Functional-Differential Inclusions in Banach Spaces, Nonlinear Analysis: Theory, Methods and Applications, Vol. 20, pp. 781–792, 1993. · Zbl 0778.34058 · doi:10.1016/0362-546X(93)90068-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.