Averaging of differential inclusions with many-valued pulses

We justify a method of complete and partial averaging on finite and infinite intervals for differential inclusions with many-valued pulses.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 11, pp. 1526–1532, November, 1995.     

Averaging of functional differential inclusions in Banach spaces

Averaging schemes for functional differential inclusions in Banach spaces with slow and fast time variables are studied. Under mild suppositions on the regularity, the periodic case and the case of non-existence of an average are investigated. The accuracy of the averaging technique is considered as well. In particular, for periodic systems, the usual linear approximation is achieved. Under stronger regularity conditions, approximation orders for Krylov-Bogoliubov-Mitropolskii type right-hand sides are derived.     

Averaging of systems of differential inclusions with slow and fast variables

We consider the system of differential inclusions

$$\dot x \in \mu F(t, x, y, \mu ), x(0) = x_0 , \dot y \in G(t, x, y, \mu ), y(0) = y_0 $$

, where F,G: D → Kυ (\(R^{m_1 } \)), Kυ (\(R^{m_2 } \)) are mappings into the sets of nonempty convex compact sets in the Euclidean spaces \(R^{m_1 } \) and \(R^{m_2 } \), respectively, D = R ₊ × \(R^{m_1 } \) × \(R^{m_2 } \) × [0, a], a > 0, and µ is a small parameter. The functions F and G and the right-hand side of the averaged problem \(\dot u\) ∈ µF ₀(u), u(0) = x ₀, F ₀(u) ∈ Kυ (\(R^{m_1 } \)), satisfy the one-sided Lipschitz condition with respect to the corresponding phase variables. Under these and some other conditions, we prove that, for each ? > 0, there exists a µ > 0 such that, for an arbitrary µ ∈ (0, µ₀] and any solution x _µ(·), y _µ(·) of the original problem, there exists a solution u _µ(·) of the averaged problem such that ∥x _µ(t) ? y _µ(t) ∥ ≤ ? for t ∈ [0, 1/µ]. Furthermore, for each solution u _µ(·)of the averaged problem, there exists a solution x _µ(·), y _µ(·) of the original problem with the same estimate.

     

Averaging of time-varying differential equations revisited

Employing comparison of integrals on a fast time scale, we offer a new criterion and simple proofs of the averaging principle for time-varying ordinary differential equations. The method allows straightforward extensions and generalizations. Comparisons with available criteria and estimates, along with examples and applications, are offered.     

Monotone trajectories of differential inclusions and functional differential inclusions with memory

The paper gives a necessary and sufficient condition for the existence of monotone trajectories to differential inclusionsdx/dt ∈S[x(t)] defined on a locally compact subsetX ofR ^p, the monotonicity being related to a given preorder onX. This result is then extended to functional differential inclusions with memory which are the multivalued case to retarded functional differential equations. We give a similar necessary and sufficient condition for the existence of trajectories which reach a given closed set at timet=0 and stay in it with the monotonicity property fort≧0.     

Averaging differential embeddings with Hukuhara derivative

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 1, pp. 121–125, January, 1989.     

Averaging for retarded functional differential equations

We consider retarded functional differential equations in the setting of Kurzweil-Henstock integrable functions and we state an averaging result for these equations. Our result generalizes previous ones.     

Differential inclusions for differential forms

We study necessary and sufficient conditions for the existence of solutions in of the problem where is a given set. Special attention is given to the case of the curl (i.e. k = 1), particularly in dimension 3. Some applications to the calculus of variations are also stated.     

Multivalued perturbations ofm-accretive differential inclusions

Given anm-accretive operatorA in a Banach spaceX and an upper semicontinuous multivalued mapF: [0,a]×X→2 ^X , we consider the initial value problemu′∈−Au+F(t,u) on [0,a],u(0)=x ₀. We concentrate on the case when the semigroup generated by—A is only equicontinuous and obtain existence of integral solutions if, in particular,X* is uniformly convex andF satisfies β(F(t,B))≤k(t)β(B) for all boundedB⊂X wherek∈L ¹([0,a]) and β denotes the Hausdorff-measure of noncompactness. Moreover, we show that the set of all solutions is a compactR _δ-set in this situation. In general, the extra condition onX* is essential as we show by an example in whichX is not uniformly smooth and the set of all solutions is not compact, but it can be omited ifA is single-valued and continuous or—A generates aC _o-semigroup of bounded linear operators. In the simpler case when—A generates a compact semigroup, we give a short proof of existence of solutions, again ifX* is uniformly (or strictly) convex. In this situation we also provide a counter-example in ℝ⁴ in which no integral solution exists. The author gratefully acknowledges financial support by DAAD within the scope of the French-German project PROCOPE.     

First-order approximations for differential inclusions

It is shown, under a mere continuity assumption, that the union of affine functions generated by the right-hand side of a differential inclusion, is a little oh approximation of the attainable set. Explicit estimates are given. An application to polygonal approximations is displayed.Research supported by a grant from the Basic Research Fund, The Israel Academy of Science and Humanities.Incumbent of the Hettie H. Heineman Professorial Chair in Mathematics.     

Optimal solutions to differential inclusions

We treat a control problem given in terms of a differential inclusion $$\dot x(t) \in E(t,x(t))$$ and develop necessary conditions for a minimum in the problem. These conditions are given in terms of certain normals to arbitrary closed sets, and require no smoothness or convexity in the problem. The results subsume related works that incorporate convexity assumptions.