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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.

MSC:
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
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References:
[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