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Strongly nonlinear impulsive evolution equations and optimal control. (English) Zbl 1065.49023
The system is described by the evolution equation \[ x'(t) + A(t, x(t)) = g(t, x(t)) + B(t)u(t) \] satisfied in an interval \(0 < t < T\) minus a finite set \(\{t_k\},\) \(0 < t_1 < t_2 < \dots < t_n < T.\) The solution satisfies an initial condition at \(t = 0\) and jump conditions at the other points, \[ x(0) = x_0, \quad x(t_k^+) - x(t_k) = F_k(x(t_k)) , \quad k = 1, 2, \dots, n . \] The author discusses existence of solutions to a control problem that consists of minimizing an integral functional \(J(x, u)\) over trajectories of the system.

49N25 Impulsive optimal control problems
49J27 Existence theories for problems in abstract spaces
34G20 Nonlinear differential equations in abstract spaces
93C25 Control/observation systems in abstract spaces
34A37 Ordinary differential equations with impulses
49J15 Existence theories for optimal control problems involving ordinary differential equations
Full Text: DOI
[1] Ahmed, N.U., Measure solution for impulsive systems in Banach spaces and their control, Dyn. continuous discrete impulsive syst., 6, 519-535, (1999) · Zbl 0951.34040
[2] Ahmed, N.U.; Teo, K.L., Optimal control of distributed parameter systems, (1981), North-Holland New York · Zbl 0472.49001
[3] Balder, E., Necessary and sufficient conditions for L1 strong-weak lower semicontinuity of integral functionals, J. nonlinear anal., 11, 1399-1404, (1987) · Zbl 0638.49004
[4] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (1980), Springer New York · Zbl 0691.35001
[5] S. Hu, N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I, Kluwer Academic Publishers, Boston, London, 1997. · Zbl 0887.47001
[6] Liu, J.H., Nonlinear impulsive evolution equations, Dyn. continuous discrete impulsive syst., 6, 77-85, (1999) · Zbl 0932.34067
[7] Rogovchenko, Y.V., Impulsive evolution systems: main result and new trends, Dyn. continuous discrete impulsive syst., 3, 77-88, (1997) · Zbl 0879.34014
[8] Sattayatham, P.; Tangmanee, S.; Wei, Wei, On periodic solution of nonlinear evolution equation, Jmaa, 276, 98-108, (2002) · Zbl 1029.34045
[9] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Vol. II, Springer, New York, 1990. · Zbl 0684.47029
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