Abstract
The paper presents a closure theorem for the attainable trajectories of a class of control systems governed by a large class of nonlinear evolution equations in reflexive Banach spaces. Several existence theorems for optimal controls are proven that include a terminal control problem, a time-optimal control problem, and a special Bolza problem. Some results of independent interest are also presented.
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Balakrishnan, A. V.,Optimal Control Problems in Banach Spaces, SIAM Journal on Control and Optimization, Vol. 3, pp. 152–180, 1965.
Conti, R.,On Some Aspects of Linear Control Theory, Mathematical Theory of Control, Edited by A. V. Balakrishnan and L. W. Neustadt, Academic Press, New, New York, 1967.
Falb, P.,Infinite Dimensional Control Problems, I, On the Closure of the Set of Attainable States for Linear Systems, Journal of Mathematical Analysis and Applications, Vol. 9, pp. 12–22, 1964.
Fattorini, H. O.,Control in Finite Time of Differential Equations in Banach Space, Communication on Pure and Applied Mathematics, Vol. 19, pp. 17–34, 1966.
Fattorini, H. O.,A Remark on the Bang-Bang Principle for Linear Control Systems in Infinite-Dimensional Space, SIAM Journal on Control and Optimization, Vol. 6, pp. 109–113, 1968.
De Julio, S.,On the Optimization of Infinite Dimensional Linear Systems, Computing Methods in Optimization Problems, 2, Edited by L. A. Zadeh, L. W. Neustadt, and A. V. Balakrishnan, Academic Press, New York, New York, 1969.
Lions, J. L.,Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, Germany, 1971.
Cole, J. K.,A Selector Theorem in Banach Spaces, Journal of Optimization Theory and Applications, Vol. 7, pp. 170–172, 1971.
Ahmed, N. U., andTeo, K. L.,Comments on a Selector Theorem in Banach Spaces, Journal of Optimization Theory and Applications, Vol. 19, pp. 117–118, 1976.
Ahmed, N. U., andTeo, K. L.,An Existence Theorem on Optimal Control of Partially Observable Diffusions, SIAM Journal on Control and Optimization, Vol. 12, pp. 351–355, 1974.
Browder, F. E.,Nonlinear Equations of Evolution, Annals of Mathematics, Vol. 80, pp. 485–523, 1964.
Browder, F. E.,Nonlinear Initial Value Problems, Annals of Mathematics, Vol. 81, pp. 51–87, 1965.
Yosida, K.,Functional Analysis, Springer-Verlag, Berlin, Germany, 1968.
Larsen, R.,Functional Analysis, Marcel Dekker, New York, New York, 1973.
Warga, J.,Optimal Control of Differential and Functional Equations, Academic Press, New York, New York, 1972.
Willard, S.,General Topology, Addison-Wesley Publishing Company, Reading, Massachusetts, 1970.
Dunford, N., andSchwartz, J. T.,Linear Operators, Part I, John Wiley and Sons, New York, New York, 1958.
Berkovitz, L. D.,Existence and Lower Closure Theorems for Abstract Control Problems, SIAM Journal on Control and Optimization, Vol. 12, pp. 27–92, 1974.
Ahmed, N. U.,Optimal Control of a Class of Strongly Nonlinear Parabolic Systems, Journal of Mathematical Analysis and Applications, Vol. 61, pp. 188–207, 1977.
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Communicated by L. Cesari
This work was supported in part by the National Research Council of Canada under Grant No. 7109.
The authors would like to thank Professor L. Cesari for pointing out that joint continuity off is required for the setsG andR to satisfy the upper semicontinuity property (Theorems 5.1 and 5.2).
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Ahmed, N.U., Teo, K.L. Optimal control of systems governed by a class of nonlinear evolution equations in a reflexive Banach space. J Optim Theory Appl 25, 57–81 (1978). https://doi.org/10.1007/BF00933255
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DOI: https://doi.org/10.1007/BF00933255