Controllability questions for nonlinear systems in abstract spaces. (English) Zbl 0697.49040
In abstract spaces, we consider certain constrained controllability and approximate controllability properties of a nonlinear system that can be reduced 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.
Reviewer: E.N.Chukwu
MSC:
| 93B05 | Controllability |
| 93C10 | Nonlinear systems in control theory |
| 93C25 | Control/observation systems in abstract spaces |
References:
| [1] | Fattorini, H. O.,The Time-Optimal Control Problem in Banach Spaces, Applied Mathematics and Optimization, Vol. 1, pp. 163-188, 1974. · Zbl 0295.49009 · doi:10.1007/BF01449028 |
| [2] | Narakawa, K.,Boundary-Value Control of Isotropic Elastodynamic Systems with Constrained Control, Journal of Mathematical Analysis Applications, Vol. 93, pp. 250-272, 1983. · Zbl 0534.73016 · doi:10.1016/0022-247X(83)90229-9 |
| [3] | 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. · Zbl 0095.29503 |
| [4] | Chukwu, E. N.,Finite Time Controllability of a Nonlinear Control Process, SIAM Journal on Control, Vol. 13, pp. 807-816, 1975. · doi:10.1137/0313047 |
| [5] | 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. · Zbl 0555.93008 · doi:10.1016/0022-247X(84)90169-0 |
| [6] | Triggiani, R.,Controllability and Observability in Banach Spaces with Bounded Operators, SIAM Journal on Control, Vol. 3, pp. 462-491, 1975. · Zbl 0268.93007 · doi:10.1137/0313028 |
| [7] | 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. · Zbl 0486.58004 · doi:10.1016/0362-546X(82)90023-2 |
| [8] | Markus, L.,Controllability of Nonlinear Processes, SIAM Journal on Control, Vol. 3, pp. 78-90, 1965. · Zbl 0294.93001 |
| [9] | Weiss, L.,On the Controllability of Delay Differential Systems, SIAM Journal on Control, Vol. 5, pp. 575-587, 1967. · Zbl 0183.16402 · doi:10.1137/0305036 |
| [10] | Underwood, R. G., andYoung, D. F.,Null Controllability of Nonlinear Functional Differential Equations, SIAM Journal on Control, Vol. 17, pp. 753-772, 1979. · Zbl 0438.93016 · doi:10.1137/0317053 |
| [11] | Wong, H. D.,Controllability for Nonlinear Differential Equations in Infinite-Dimensional Spaces, PhD Thesis, University of Minnesota, 1979. |
| [12] | 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. · Zbl 0549.49028 · doi:10.1007/BF00940766 |
| [13] | Ladas, G. E., andLakshmikantham, V.,Differential Equations in Abstract Spaces, Academic Press, New York, New York, 1972. · Zbl 0257.34002 |
| [14] | Lang, S.,Analysis, Vol. 2, Addison-Wesley, Reading, Massachusetts, 1969. · Zbl 0176.00504 |
| [15] | Hale, J.,Theory of Functional Differential Equations, Springer-Verlag, New York, New York, 1977. · Zbl 0352.34001 |
| [16] | Dieudonne, J.,Foundations of Modern Analysis, Academic Press, New York, New York, 1969. |
| [17] | Chukwu, E. N.,Global Behavior of Retarded Functional Differential Equations. Preprint, North Carolina State University, 1989. · Zbl 0721.34086 |
| [18] | Salamon, D.,Control and Observation of Neutral Systems, Pitman Publishing, Boston, Massachusetts, 1984. · Zbl 0546.93041 |
| [19] | 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. · Zbl 0533.34052 |
| [20] | 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. · Zbl 0469.93016 · doi:10.1137/0319050 |
| [21] | 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. · Zbl 0354.93014 · doi:10.1137/0315028 |
| [22] | 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. · Zbl 0305.93013 · doi:10.1016/0022-247X(75)90033-5 |
| [23] | 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. · Zbl 0397.93001 · doi:10.1137/1020095 |
| [24] | Pazy, A.,Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, New York, 1983. · Zbl 0516.47023 |
| [25] | 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. · Zbl 0732.92029 |
| [26] | 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. · Zbl 0596.49006 |
| [27] | 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. · Zbl 0614.34067 |
| [28] | Triggiani, R.,Controllability, Observability, and Stabilizability of Dynamical Systems in Banach Space with Bounded Operators, PhD Thesis, University of Michigan, 1973. |
| [29] | Nakagiri, S.,Optimal Control of Linear Retarded Systems in Banach Spaces, Journal of Mathematical Analysis and Applications, Vol. 120, pp. 169-210, 1986. · Zbl 0603.49005 · doi:10.1016/0022-247X(86)90210-6 |
| [30] | 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. · Zbl 0426.93012 · doi:10.1109/TAC.1980.1102241 |
| [31] | Curtain, R. F., andPritchard, A. J.,Infinite-Dimensional Linear System Theory, Springer-Verlag, New York, New York, 1978. · Zbl 0389.93001 |
| [32] | Ladas, G. E., andLakshmikantham, V.,Differential Equations in Abstract Spaces, Academic Press, New York, New York, 1972. · Zbl 0257.34002 |
| [33] | Chukwu, E. N.,Control of Linear Systems, Preprint, North Carolina State University, 1990. |
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.
