On Differential Inclusions with Predetermined Attainable Sets
by
Kh.G. Guseinov
Department of Mathematics, Anadolu University, 26470 Eskisehir, TURKEY
Coauthors: Orhan Ozer, Serkan Duzce

The attainable set of the given differential inclusion (DI) at the given instant of time, is one of the important constructions of the DI theory. Numerous papers have been devoted to study various topological properties of the attainable sets of the given DI and there are different methods for their numerical calculation and estimation (see, e.g. [1, 4, 6-8, 12-15] and references therein). Consider the DI

× x   in F( t, x)

(\theequation)

where x in Rⁿ- is the phase state vector, t in [t₀, \theta] is the time. By symbol X(t _* , X _* ) we denote the set of all solutions of the DI (1) satisfying the condition x( t _* ) in X _* , where X _* subset Rⁿ,  t _* in [t₀, \theta] . We set

X( t;t * , X * ) = { x( t) in Rn:x( ·) in X(t * , X * ) }

and denote by symbol h( A, C) the Hausdorff distance between the sets A,  C subset Rⁿ. The set X(t;t _* , X _* ) is called the attainable set of the DI (1) at the moment of the time t with initial set (t _* , X _* ) .

In this paper, an inverse problem of the DI theory is considered. For a given \epsilon > 0 and a given compact convex valued and continuous set valued map t --> Z( t) , t in [ t₀, \theta] , it is required to specify a DI (1) such that the inequality

h( X( t;t0, Z(t0)) , Z(t) ) <= \epsilon

would be held for every t in [ t₀, \theta] . Such problems may appear in mathematics modelling where it is required to specify the dynamic of the system through measurement of the phase state of the system.

The inverse problem was investigated in the works [3, 5, 9, 10]. In this study, the desired DI is defined so that the right hand side of the DI satisfies the conditions, which guarantee existence and extendability of the solutions. The solution of the problem is based on the existence of the convex extension of the special type convex compact set valued map. Note that the notions strong and weak invariant sets with respect to DI play an important role in construction of such DI (see., e.g. [1, 2, 7, 8, 11]).

To solve the problem, we define the set valued map t --> W( t) , t in [ t₀, \theta] , where W( t) is \frac \epsilon4 neighborhood of the set Z( t). Then, we take a small enough partition of the closed interval [ t₀, \theta] and on each of subintervals of the partition, we specify affine interpolation of the compact convex valued and continuous set valued map t --> W( t) , t in [ t₀, \theta] , where int  W(t) =/= \emptyset for every t in [ t₀, \theta] and int  W(t) means the interior of W(t). Using such approximated affine tubes, we construct a linear DI, which solves the problem.

References

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Date received: April 20, 2004

Copyright © 2004 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caoe-22.