Abstract
In abstract spaces, we consider certain constrained controllability and approximate controllability properties of a nonlinear system that can be deduced from various controllability properties of its associated linear system. Several examples involving partial differential operators and functional delay operators are given to illustrate the theory.
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References
Fattorini, H. O.,The Time-Optimal Control Problem in Banach Spaces, Applied Mathematics and Optimization, Vol. 1, pp. 163–188, 1974.
Narakawa, K.,Boundary-Value Control of Isotropic Elastodynamic Systems with Constrained Control, Journal of Mathematical Analysis Applications, Vol. 93, pp. 250–272, 1983.
LaSalle, J.,The Time-Optimal Control Problem, The Theory of Nonlinear Oscillations, Edited by A. A. Andronov and C. E. Chaikin, Princeton University Press, Princeton, New Jersey, Vol. 5, pp. 1–24, 1959.
Chukwu, E. N.,Finite Time Controllability of a Nonlinear Control Process, SIAM Journal on Control, Vol. 13, pp. 807–816, 1975.
Chukwu, E. N.,Null Controllability in Function Space of Nonlinear Retarded Systems with Limited Control, Journal of Mathematical Analysis and Applications, Vol. 103, pp. 198–210, 1984.
Triggiani, R.,Controllability and Observability in Banach Spaces with Bounded Operators, SIAM Journal on Control, Vol. 3, pp. 462–491, 1975.
Craven, B. D., andNashed, M. Z.,Generalized Implicit Function Theorems When the Derivative Has No Bounded Inverse, Journal of Nonlinear Analysis, Vol. 6, pp. 375–387, 1982.
Markus, L.,Controllability of Nonlinear Processes, SIAM Journal on Control, Vol. 3, pp. 78–90, 1965.
Weiss, L.,On the Controllability of Delay Differential Systems, SIAM Journal on Control, Vol. 5, pp. 575–587, 1967.
Underwood, R. G., andYoung, D. F.,Null Controllability of Nonlinear Functional Differential Equations, SIAM Journal on Control, Vol. 17, pp. 753–772, 1979.
Wong, H. D.,Controllability for Nonlinear Differential Equations in Infinite-Dimensional Spaces, PhD Thesis, University of Minnesota, 1979.
Ahmed, N. U.,Finite-Time Controllability for a Class of Linear Evolution Equations in a Banach Space with Control Constraints, Journal of Optimization Theory and Applications, Vol. 47, pp. 129–158, 1985.
Ladas, G. E., andLakshmikantham, V.,Differential Equations in Abstract Spaces, Academic Press, New York, New York, 1972.
Lang, S.,Analysis, Vol. 2, Addison-Wesley, Reading, Massachusetts, 1969.
Hale, J.,Theory of Functional Differential Equations, Springer-Verlag, New York, New York, 1977.
Dieudonne, J.,Foundations of Modern Analysis, Academic Press, New York, New York, 1969.
Chukwu, E. N.,Global Behavior of Retarded Functional Differential Equations. Preprint, North Carolina State University, 1989.
Salamon, D.,Control and Observation of Neutral Systems, Pitman Publishing, Boston, Massachusetts, 1984.
Chukwu, E.,An Estimate for the Solution of Certain Functional Differential Equations of Neutral Type, Nonlinear Phenomena in Mathematical Sciences. Edited by V. Lakshmikantham, Academic Press, New York, New York, 1982.
Chen, G., Mills, W. H., andCrosta, G.,Exact Controllability Theorems and Numerical Simulations for Some Nonlinear Differential Equations, SIAM Journal on Control, Vol. 19, pp. 765–780, 1981.
Triggiani, R.,A Note on the Lack of Exact Controllability for Mild Solutions in Banach Spaces, SIAM Journal on Control, Vol. 15, pp. 407–411, 1977.
Triggiani, R.,On the Lack of Exact Controllability for Mild Solutions in Banach Spaces, Journal of Mathematical Analysis and Applications, Vol. 50, pp. 438–446, 1975.
Russell, D. L.,Controllability and Stability Theory for Linear Partial Differential Equations: Recent Progress and Open Questions, SIAM Review, Vol. 20, pp. 639–739, 1978.
Pazy, A.,Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, New York, 1983.
Chukwu, E. N.,The Time-Optimal Control of Nonlinear Delay Equations, Operator Methods for Optimal Control Problems, Edited by S. J. Lee, Marcel Dekker, New York, New York, 1988.
Papageorgiou, N. S.,Existence of Optimal Controls for Nonlinear Systems in Banach Spaces, Journal of Optimization Theory and Applications, Vol. 53, pp. 481–489, 1987.
Chukwu, E. N.,The Time Optimal Control Theory of Nonlinear Systems of Neutral Type, Nonlinear Analysis and Applications, Edited by V. Lakshmikantham, Marcel Dekker, New York, New York, 1987.
Triggiani, R.,Controllability, Observability, and Stabilizability of Dynamical Systems in Banach Space with Bounded Operators, PhD Thesis, University of Michigan, 1973.
Nakagiri, S.,Optimal Control of Linear Retarded Systems in Banach Spaces, Journal of Mathematical Analysis and Applications, Vol. 120, pp. 169–210, 1986.
Barmish, B. R., andSchmitendorf, W. E.,A Necessary and Sufficient Condition for Local Constrained Controllability of Linear Systems, IEEE Transactions on Automatic Control, Vol. AC-25, pp. 97–100, 1980.
Curtain, R. F., andPritchard, A. J.,Infinite-Dimensional Linear System Theory, Springer-Verlag, New York, New York, 1978.
Ladas, G. E., andLakshmikantham, V.,Differential Equations in Abstract Spaces, Academic Press, New York, New York, 1972.
Chukwu, E. N.,Control of Linear Systems, Preprint, North Carolina State University, 1990.
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Communicated by W. E. Schmitendorf
The research of the second author was supported in part by NSF Grant DMS-85-088651 and by the University of Tennessee Science Alliance Award.
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Chukwu, E.N., Lenhart, S.M. Controllability questions for nonlinear systems in abstract spaces. J Optim Theory Appl 68, 437–462 (1991). https://doi.org/10.1007/BF00940064
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DOI: https://doi.org/10.1007/BF00940064