Analysis and control of second-order differential-operator inclusions with +-coercive damping
- PDF / 135,347 Bytes
- 9 Pages / 595.276 x 793.701 pts Page_size
- 45 Downloads / 189 Views
ANALYSIS AND CONTROL OF SECOND-ORDER DIFFERENTIAL-OPERATOR INCLUSIONS WITH +-COERCIVE DAMPING N. V. Zadoyanchuka† and P. O. Kasyanov a‡
UDC 517.9
Second-order differential-operator inclusions with weakly coercive pseudomonotone mappings are considered. Function-topological properties of a resolving operator are investigated. The results are applied to mathematical models of the nonlinearized theory of viscoelasticity. Keywords: second-order differential inclusion, +-coercive operator, optimal control, viscoelasticity.
INTRODUCTION In investigating mathematical models of nonlinear processes and fields of the nonlinearized theory of viscoelasticity and piezoelectrics and studying waves of different nature, the following scheme is often used: such a model is reduced to some differential-operator inclusion or a multivariational inequality in an infinite-dimensional space [1–3]. Next, using some method of approximation or other, the existence of a generalized solution of this problem is proved, constructive methods of searching for approximate solutions are substantiated, and functional-topological properties of the resolving operator [2] are studied. If the character of the mentioned process is evolutionary, its mathematical model is described with the help of a second-order differential-operator inclusion [4–6]. In this case, relationships between key parameters of the original problem provide definite properties for a multivalued (in the general case) mapping in the differential-operator scheme of investigation. We note that, in the majority of works devoted to this line of investigation, rather stringent conditions connected with uniform coercitivity, boundedness, and generalized pseudo-monotonicity are imposed on “damping” [4, 6]. As a rule, these conditions not only provide the existence of solutions for such problems but also guarantee the dissipation of all solutions and sometimes the existence of a global compact attractor, which does not always naturally reflect the actual behavior of the geophysical process or field being considered [7]. Therefore, the need arises for the investigation of functional-topological properties of the resolving operator for a differential-operator inclusion that describe, in particular, new wider classes of nonlinear processes and fields of the nonlinearized theory of viscoelasticity with an adequate essential weakening of the above-mentioned properties of differential operators with corresponding applications to concrete mathematical models. In this work, problems of analysis and control of second-order differential-operator inclusions with weakly coercive w l0 -pseudo-monotone mappings are considered. The dependence of solutions on functional parameters of a problem is investigated, and an optimal control problem is considered. The results obtained are applied to mathematical models of the nonlinearized theory of viscoelasticity. PROBLEM STATEMENT Let V0 and Z 0 be real reflexive separable Banach spaces with the corresponding norms || × ||V0 and || × || Z 0 , and let H 0
be a
Data Loading...
