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Systems governed by impulsive differential inclusions on Hilbert spaces. (English) Zbl 0995.34053
The author considers systems in a Hilbert space governed by impulsive differential inclusions of the form $$\dot x(t)\in Ax(t)+F\bigl(x(t) \bigr),\ t\in[0,T] \setminus D,\ x(0)=x_0,$$ $$\Delta x(t_i)\in G_i\bigl(x(t_i) \bigr), \ 0=t_0<t_1 <\dots<t_{n+1}=T,$$ where $A$ is the infinitesimal generator of a $C_0$-semigroup, $F$ and the $G_i$’s are multivalued, $D=\{t_1,t_2, \dots, t_n\} \subseteq(0,T)$ and $\Delta x(t_i)= x(t_i+0)-x(t_i)$. The $G_i$’s being set-valued allow for systems in which the jump sizes are uncertain and also problems in which the jump sizes are chosen from a control set. The existence of solutions is proven under the assumptions $F$ is closed convex-valued, satisfies a growth condition and is upper semicontinuous; the $G_i$’s are closed bounded-valued and map closed and bounded sets into closed and bounded sets; plus several other conditions. Under the additional assumption that the semigroup generated by $A$ is compact, the attainable set is characterized. Special cases of this problem in which $F$ or the $G_i$’s are singleton-valued are also considered. Finally, the author mentions several open questions in control theory relating to these results.

34G25Evolution inclusions
34H05ODE in connection with control problems
34A37Differential equations with impulses
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
49N25Impulsive optimal control problems
93B03Attainable sets
Full Text: DOI
[1] N.U. Ahmed, Optimal impulse control for impulsive systems in Banach spaces, Int. J. Differential Equations Appl., to appear. · Zbl 0959.49023
[2] N.U. Ahmed, Measure solutions for impulsive systems in Banach spaces and their control, J. Dyn. Cont. Discrete Impulsive Systems, 6 (1999), 519--535. · Zbl 0951.34040
[3] Ahmed, N. U.: Measure solutions for semilinear evolution equations with polynomial growth and their optimal control. Discussiones mathematicae -- differential inclusions 17, 5-27 (1997) · Zbl 0905.34056
[4] Ahmed, N. U.: Optimal control of infinite dimensional systems governed by functional differential inclusions. Discussiones mathematicae -- differential inclusions 15, 75-94 (1995) · Zbl 0824.49007
[5] N.U. Ahmed, Optimization and identification of systems governed by evolution equations on Banach space, Pitman Research Notes in Mathematics Series, Vol. 184, 1988, Longman Scientific and Technical, U.K., Co-published with Wiley, New York. · Zbl 0645.93001
[6] N.U. Ahmed, Semigroup theory with applications to systems and control, Pitman Research Notes in Mathematics Series, 246, Longman Scientific and Technical, U.K., Co-published with Wiley, New York, 1991. · Zbl 0727.47026
[7] V. Barbu, Optimal control of variational inequalities, Pitman Research Notes in Mathematics, Vol. 100, Boston, 1984. · Zbl 0574.49005
[8] Deimling, K.: Fixed points of weakly inward of multivalued maps. Nonlinear anal. TMA 10, 465-469 (1986) · Zbl 0607.47055
[9] Deimling, K.: Multivalued differential inclusions, Walter de gruyter, Berlin, New York, 1992. Nonlinear anal. TMA 10, 465-469 (1986) · Zbl 0607.47055
[10] H.O. Fattorini, Infinite dimensional optimization and control theory, Encyclopedia of Mathematics and its applications, Cambridge University Press, Cambridge, 1998.
[11] Guo, D.; Liu, X. Z.: First order integro-differential equations on unbounded domain in a Banach space. Dyn. cont. Discrete impulsive systems 2, No. 3, 77-88 (1996)
[12] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. · Zbl 0719.34002
[13] Liu, J. H.: Nonlinear impulsive evolution equations. Dyn. cont. Discrete impulsive systems 6, No. 1, 77-85 (1999) · Zbl 0932.34067
[14] Rogovchenko, Y. V.: Impulsive evolution systems: Main results and new trends. Dyn. cont. Discrete impulsive systems 3, No. 1, 77-88 (1997) · Zbl 0879.34014