Evolution Inclusions Research Articles

Existence and approximate controllability for systems governed by fractional delay evolution inclusions

We prove the existence of mild solutions for the control system governed by fractional delay evolution inclusion in a Banach space X:assuming that the operator A generates a strongly continuous semigroup and the nonlinear part F is a multi-valued function with convex and closed values for which is weakly upper semicontinuous for a.e. and has a -integrable selection for each . Moreover, in this setting, we obtain the approximate controllability for such system provided that the corresponding linear control system is approximately controllable. Finally, our results apply, in particular, to the control problem of fractional partial differential inclusion with time delay.

On the ‘bang-bang’ principle for a class of fractional semilinear evolution inclusions

In this paper, we study the problem of the existence of solutions for a class of fractional semilinear evolution inclusions with non-convex right-hand side. We mainly focus on the existence of the extreme solution and the relationship of the solution sets between the original problem and the convexified problem. An example is provided to illustrate the application of the obtained theory.

Controllability for systems governed by semilinear evolution inclusions without compactness

In this paper we study the controllability for a class of semilinear differential inclusions in Banach spaces. Since we assume the regularity of the nonlinear part with respect to the weak topology, we do not require the compactness of the evolution operator generated by the linear part. As well we are not posing any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. We are considering the usual assumption on the controllability of the associated linear problem. Notice that, in infinite dimensional spaces, the above mentioned compactness of the evolution operator and linear controllability condition are in contradiction with each other. We suppose that the nonlinear term has convex, closed, bounded values and a weakly sequentially closed graph when restricted to its second argument. This regularity setting allows us to solve controllability problem under various growth conditions. As application, a controllability result for hyperbolic integro-differential equations and inclusions is obtained. In particular, we consider controllability of a system arising in a model of nonlocal spatial population dispersal and a system governed by the second order one-dimensional telegraph equation.

On integro‐differential inclusions with operator‐valued kernels

AbstractWe study integro‐differential inclusions in Hilbert spaces with operator‐valued kernels and give sufficient conditions for the well‐posedness. We show that several types of integro‐differential equations and inclusions are covered by the class of evolutionary inclusions, and we therefore give criteria for the well‐posedness within this framework. As an example, we apply our results to the equations of visco‐elasticity and to a class of nonlinear integro‐differential inclusions describing phase transition phenomena in materials with memory. Copyright © 2014 John Wiley & Sons, Ltd.

Anti-periodic extremal problems for a class of nonlinear evolution inclusions in R N

The aim of this paper is to establish the existence of anti-periodic solutions to the following nonlinear anti-periodic problem: a.e. , , in where denotes the extremal point set of the multifunction , and is a nonlinear map from to . Sufficient conditions for the existence of extremal solutions are presented. Also, we prove that the extremal point set of this problem is compact in and dense in the solution set of nonlinear evolution problems with a convex valued perturbation which is multivalued. We apply our results on the control system with a priori feedback.

Optimal Control of Semilinear Unbounded Evolution Inclusions with Functional Constraints

This paper is devoted to the study of a Mayer-type optimal control problem for semilinear unbounded evolution inclusions in reflexive and separable Banach spaces subject to endpoint constraints described by finitely many Lipschitzian equalities and inequalities. First we construct a sequence of discrete approximations to the optimal control problem for evolution inclusions and prove that optimal solutions to discrete approximation problems uniformly converge to a given optimal solution for the original continuous-time problem. Then, based on advanced tools of variational analysis and generalized differentiation, we derive necessary optimality conditions for discrete-time problems under fairly general assumptions. Combining these results with recent achievements of variational analysis in infinite-dimensional spaces, we establish new necessary optimality conditions for continuous-time evolution inclusions by passing to the limit from discrete approximations.

Well-posedness of non-autonomous evolutionary inclusions

A class of non-autonomous differential inclusions in a Hilbert space setting is considered. The well-posedness for this class is shown by establishing the mappings involved as maximal monotone relations. Moreover, the causality of the so established solution operator is addressed. The results are exemplified by the equations of thermoplasticity with time dependent coefficients and by a non-autonomous version of the equations of viscoplasticity with internal variables.

An extension of the Fitzpatrick theory

In the seminal work [MR 1009594], Fitzpatrick proved that for any maximal monotone operator $\alpha: V\to {\mathcal P}(V')$ ($V$ being a real Banach space) there exists a lower semicontinuous, convex representative function $f_\alpha: V \times V'\to R\cup \{+\infty\}$ such that \begin{eqnarray} f_\alpha(v,v') \ge \langle v',v\rangle \quad\;\forall (v,v'), \qquad\quad f_\alpha(v,v') = \langle v',v\rangle \;\;\Leftrightarrow\;\;\; v'\in \alpha(v). \end{eqnarray} Here we assume that $\alpha_v$ is a maximal monotone operator for any $v\in V$, and extend the Fitzpatrick theory to provide a new variational formulation for either stationary or evolutionary (nonmonotone) inclusions of the form $\alpha_v(v) \ni v'$. For any $v'\in V'$, we prove existence of a solution via the classical minimax theorem of Ky Fan. Applications include stationary and evolutionary pseudo-monotone operators, and variational inequalities.

Existence of anti-periodic solutions for a class of first order evolution inclusions

The existence of anti-periodic solutions for a class of first order nonlinear evolution inclusions defined in the framework of an evolution triple of spaces is considered. We study the problems under both convexity and nonconvexity conditions on the multivalued right-hand side. The main tools in our study are the maximal monotone property of the derivative operator with anti-periodic conditions, the surjectivity result for L-pseudomonotone operators and continuous extreme selection results from multivalued analysis. An example of a nonlinear parabolic problem is given to illustrate our results.

Multi-valued, singular stochastic evolution inclusions

We provide an abstract variational existence and uniqueness result for multi-valued, monotone, non-coercive stochastic evolution inclusions in Hilbert spaces with general additive and Wiener multiplicative noise. As examples we discuss certain singular diffusion equations such as the stochastic 1-Laplacian evolution (total variation flow) in all space dimensions and the stochastic singular fast-diffusion equation. In case of additive Wiener noise we prove the existence of a unique weak-⁎ mean ergodic invariant measure.

Existence of Solutions for a Class of Nonlinear Evolution Inclusions with Nonlocal Conditions

This paper deals with the nonlocal problems for a class of nonlinear first-order evolution inclusions. Some existence results are established for the cases of a convex and of a nonconvex valued perturbation terms. Also, the existence of extremal solutions and a strong relaxation theorem are obtained. Subsequently a nonlinear hyperbolic optimal control problem is considered and the existence theorems based on the proven results are obtained. Then the nonlinear version of "bang---bang" principle for control systems is given as well by utilizing the relaxation theorem.

EXISTENCE OF SOLUTIONS FOR IMPULSIVE PARTIAL NEUTRAL FUNCTIONAL EVOLUTION INTEGRODIFFERENTIAL INCLUSIONS WITH INFINITE DELAY

This paper investigates a class of impulsive partial neutral func- tional integrodifferential evolution inclusions with infinite delay in Banach spaces. The existence of mild solutions of these inclusions is determined under the mixed Lipschitz and Caratheodory conditions by using another nonlinear alternative of Leray-Schauder type for multivalued maps due to D. O'Regan. At the end, one example is presented.

Nonlinear evolution inclusions with one-sided perron right-hand side

In a Banach space X with uniformly convex dual, we study the evolution inclusion of the form $$ {x}^{\prime}(t)\in Ax(t)+F\left( {x(t)} \right) $$ , where A is an m-dissipative operator and F is an upper hemicontinuous multifunction with nonempty convex and weakly compact values. If X* is uniformly convex and F is one-sided Perron with sublinear growth, then, we prove a variant of the well known Filippov-Pliś theorem. Afterward, sufficient conditions for near viability and (strong) invariance of a set $$ K\subseteq \overline{D(A)} $$ are established. As applications, we derive ? - ? lower semicontinuity of the solution map and, consequently, the propagation of continuity of the minimum time function associated with the null controllability problem.

Accuracy of discrete schemes for a class of abstract evolution inequalities

For a triple of Hilbert spaces {V, H, V*}, we study a discrete and a semidiscrete scheme for an evolution inclusion of the form u′(t) + A(t)u(t) + ∂ϕ(t, u(t)) ∋ f(t), u(0) = u 0, t ∈ (0, T], where the pair {A(t), ϕ(t, ·)} consists of a family of nonlinear operators from V into V* and a family of proper convex lower semicontinuous functionals with common effective domain D(ϕ) ⊂ V. The discrete scheme is a combination of the Galerkin method with perturbations and the implicit Euler method. Under conditions on the data providing the existence and uniqueness of the solution of the problem in the space H 1(0, T; V) ∩ W ∞ 1 (0, T;H), we obtain an abstract estimate for the method error in the energy norm of first-order accuracy with respect to the time increment. By way of application, we consider a problem with an obstacle inside the domain, for which we obtain an optimal estimate of the accuracy of two implicit schemes (standard and new) on the basis of the finite element method.

Asymptotic behavior analysis on multivalued evolution inclusion with projection in Hilbert space

Let be a real Hilbert space, a closed convex subset of , and continuous convex and bounded from below on . We study the asymptotic behaviour of the non-autonomous differential inclusion with project operator as follows where is an absolutely continuous decreasing control function with strictly positive value and . On the one hand, when , each solution of PS with initial point (not necessarily in ) converges to an element of in the weak topology. On the other hand, when and , the weak and strong convergence of PS to the minimizer of over can be guaranteed under some conditions. Strong convergence can be obtained when or is strongly convex on . Another two sufficient conditions are given to ensure the weak convergence. Moreover, asymptotic analysis for the special case that is also considered. Finally, we present some discussion on the global existence of the solutions of PS.

Nonlinear evolution inclusions: Topological characterizations of solution sets and applications

This paper deals with a nonlinear delay differential inclusion of evolution type involving m-dissipative operator and source term of multi-valued type in a Banach space. Under rather mild conditions, the Rδ-structure of C0-solution set is studied on compact intervals, which is then used to obtain the Rδ-property on non-compact intervals. Secondly, the result about the structure is furthermore employed to show the existence of C0-solutions for the inclusion (mentioned above) subject to nonlocal condition defined on right half-line. No nonexpansive condition on nonlocal function is needed. As samples of applications, we consider a partial differential inclusion with time delay and then with nonlocal condition at the end of the paper.

Approximate controllability of fractional functional evolution inclusions with delay in Hilbert spaces

Approximate controllability of fractional functional evolution inclusions with delay in Hilbert spaces

AUTONOMOUS EVOLUTIONARY INCLUSIONS WITH APPLICATIONS TO PROBLEMS WITH NONLINEAR BOUNDARY CONDITIONS

We study an abstract class of autonomous differential inclusions in Hilbert spaces and show the well-posedness and causality, by establishing the operators involved as maximal monotone operators in time and space. Then the proof of the well-posedness relies on a well-known perturbation result for maximal monotone operators. Moreover, we show that certain types of nonlinear boundary value problems are covered by this class of inclusions and we derive necessary conditions on the operators on the boundary in order to apply the solution theory. We exemplify our findings by two examples.

Non-autonomous evolution inclusions with nonlocal history conditions: Global integral solutions

This paper is concerned with nonlinear nonlocal differential inclusion of evolution type in Fréchet spaces, defined on right half-line. The underlying feature of the inclusion under consideration is that it is non-autonomous. We obtain some compactness characterizations of integral solution sets for the inclusion without nonlinear perturbations. Then, making use of these characterizations, we derive a new existence result of global integral solutions for the original inclusion. No invariance condition on the nonlinearity is involved. The results we obtained here extend the semilinear case of the previous related ones such as [I.I. Vrabie, Existence in the large for nonlinear delay evolution inclusions with nonlocal initial conditions, J. Funct. Anal. 262 (2012) 1363–1391] and are new even for the case of the nonlinearity being a single-valued function.

Properties of the solutions set for a class of nonlinear evolution inclusions with nonlocal conditions

In this paper, we consider the nonlocal problems for nonlinear first-order evolution inclusions in an evolution triple of spaces. Using techniques from multivalued analysis and fixed point theorems, we prove existence theorems of solutions for the cases of a convex and of a nonconvex valued perturbation term with nonlocal conditions. Also, we prove the existence of extremal solutions and a strong relaxation theorem. Some examples are presented to illustrate the results. MSC:34B15, 34B16, 37J40.