Second-order Differential Inclusions Research Articles

Mild Solutions of Second-Order Semilinear Impulsive Differential Inclusions in Banach Spaces

In this paper, the existence of a mild solution to the Cauchy problem for impulsive semilinear second-order differential inclusion in a Banach space is investigated in the case when the nonlinear term also depends on the first derivative. This purpose is achieved by combining the Kakutani fixed point theorem with the approximation solvability method and the weak topology. This combination enables obtaining the result under easily verifiable and not restrictive conditions on the impulsive terms, the cosine family generated by the linear operator and the right-hand side while avoiding any requirement for compactness. Firstly, the problems without impulses are investigated, and then their solutions are glued together to construct the solution to the impulsive problem step by step. The paper concludes with an application of the obtained results to the generalized telegraph equation with a Balakrishnan–Taylor-type damping term.

Existence of solutions for a Lipschitzian vibroimpact problem with time-dependent constraints

We study a mechanical system with a finite number of degrees of freedom, subjected to perfect time-dependent frictionless unilateral (possibly nonconvex) constraints with inelastic collisions on active constraints. The dynamic is described in the form of a second-order measure differential inclusion. Under some regularity assumptions on the data, we establish several properties of the set of admissible positions, which is not necessarily convex but assumed to be uniformly prox-regular. Our approach does not require any second-order information or boundedness of the Hessians of the constraints involved in the problem and are specific to moving sets represented by inequalities constraints. On that basis, we are able to discretize our problem by the time-stepping algorithm and construct a sequence of approximate solutions. It is shown that this sequence possesses a subsequence converging to a solution of the initial problem. This methodology is not only used to prove an existence result but could be also used to solve numerically the vibroimpact problem with time-dependent nonconvex constraints.

Bound Sets Approach to Impulsive Floquet Problems for Vector Second-Order Differential Inclusions

In this paper, the existence and the localization of a solution of an impulsive vector multivalued second-order Floquet boundary value problem are investigated. The method used in the paper is based on the combination of a fixed point index technique with bound sets approach. At first, problems with upper-Carathéodory right-hand sides are investigated and it is shown afterwards how can the conditions be simplified in more regular case of upper semi-continuous right hand side. In this more regular case, the conditions ensuring the existence and the localization of a solution are put directly on the boundary of the considered bound set. This strict localization of the sufficient conditions is very significant since it allows some solutions to escape from the set of candidate solutions. In both cases, the \(C^1\)-bounding functions with locally Lipschitzian gradients are considered at first and it is shown afterwards how the conditions change in case of \(C^2\)-bounding functions. The paper concludes with an application of obtained results to Liénard-type equations and inclusions and the comparisons of our conclusions with the few results related to impulsive periodic and antiperiodic Liénard equations are obtained.

Well-posed control problems related to second-order differential inclusions

<p style='text-indent:20px;'>The paper deals with quadratic optimal control problems, we study the equivalence between well-posed problems and affinity on the control for a second-order differential inclusions with two-points conditions, governed by a maximal monotone operator in a finite dimensional space.</p>

Optimal feedback control for a class of second-order evolution differential inclusions with Clarke’s subdifferential

Optimal feedback control for a class of second-order evolution differential inclusions with Clarke’s subdifferential

On a second-order differential inclusion with certain integral and multi-strip boundary conditions

We study a second-order differential inclusion with integral and multi-strip boundary conditions defined by a set-valued map with nonconvex values. We obtain an existence result and we prove the arcwise connectedness of the solution set of the considered problem.

Accelerated differential inclusion for convex optimization

This paper introduces a second-order differential inclusion for unconstrained convex optimization. In continuous level, solution existence in proper sense is obtained and exponential decay of a novel Lyapunov function along with the solution trajectory is derived as well. Then in discrete level, based on numerical discretizations of the continuous model, two inexact proximal point algorithms are proposed, and some new convergence rates are established via a discrete Lyapunov function.

Optimization of Mayer functional in problems with discrete and differential inclusions and viability constraints

This paper derives the optimality conditions for a Mayer problem with discrete and differential inclusions with viable constraints. Applying necessary and sufficient conditions of problems with geometric constraints, we prove optimality conditions for second order discrete inclusions. Using locally adjoint mapping, we derive Euler-Lagrange form conditions and transversality conditions for the optimality of the discrete approximation problem. Passing to the limit, we establish sufficient conditions to the optimal problem with viable constraints. Conditions ensuring the existence of solutions to the viability problems for differential inclusions of second order have been studied in recent years. However, optimization problems of second-order differential inclusions with viable constraints considered in this paper have not been examined yet. The results presented here are motivated by practices for optimization of various fields as the mass movement model well known in traffic balance and operations research.

The optimality principle for second-order discrete and discrete-approximate inclusions

This paper deals with the necessary and sufficient conditions of optimality for the Mayer problem of second-order discrete and discrete-approximate inclusions. The main problem is to establish the approximation of second-order viability problems for differential inclusions with endpoint constraints. Thus, as a supplementary problem, we study the discrete approximation problem and give the optimality conditions incorporating the Euler-Lagrange inclusions and distinctive transversality conditions. Locally adjoint mappings (LAM) and equivalence theorems are the fundamental principles of achieving these optimal conditions, one of the most characteristic properties of such approaches with second-order differential inclusions that are specific to the existence of LAMs equivalence relations. Also, a discrete linear model and an example of second-order discrete inclusions in which a set-valued mapping is described by a nonlinear inequality show the applications of these results.

Infinitely Many Solutions for Second-Order Impulsive Differential Inclusions with Relativistic Operator

In this paper, the boundary value problem of second-order impulsive differential inclusion involving relativistic operator is studied. Infinitely many nonnegative solutions are obtained by using non-smooth critical point theorem for locally Lipschitz functionals.

Duality Problems with Second-Order Polyhedral Discrete and Differential Inclusions

The present paper deals with the theory of duality for the Mayer problem given by second-order polyhedral discrete and differential inclusions. First, we formulate the conditions of optimality in the form of Euler–Lagrange type inclusions and transversality conditions for the polyhedral problems with second-order discrete and differential inclusions. Second, we establish dual problems for discrete and differential inclusions based on the infimal convolution concept of convex functions and prove the results of duality. For both primary and dual problems, the Euler–Lagrange type inclusions are “duality relations” and that the dual problem for discrete-approximate problems bridges the gap between the dual problems of discrete and continuous problems. As a result, the passage to the limit in the dual problem with discrete approximations plays a substantial role in the following investigations, without any which can hardly ever be established by any duality to the continuous problem. Furthermore, the numerical results also are provided.

Polyhedral optimization of second-order discrete and differential inclusions with delay

Polyhedral optimization of second-order discrete and differential inclusions with delay

Existence results for second-order nonlinear differential inclusion with nonlocal boundary conditions

Existence results for second-order nonlinear differential inclusion with nonlocal boundary conditions

Existence Results for Second-Order Nonlinear Differential Inclusion with Nonlocal Boundary Conditions

Existence Results for Second-Order Nonlinear Differential Inclusion with Nonlocal Boundary Conditions

Systems of simultaneous differential inequalities, inclusions and subordinations in the complex plane

There are many articles in the literature dealing with a first, second and third-order differential inequalities, inclusions or subordinations in the complex plane, none of which deals with systems of such topics. This article investigates systems of two second-order simultaneous differential inequalities, inclusions and subordinations in two complex functions p and q in the complex plane. A typical example of a system of differential inequalities is $$ \left\{ {\begin{array}{*{20}l} {\text{Re} [2p(z) + zp^{{\prime }} (z) + z^{2} p^{{\prime \prime }} (z) - q(z)] > 0,} \hfill \\ {\text{Re} [p(z) + 7zq^{{\prime }} (z)] > 4.} \hfill \\ \end{array} } \right. $$ The authors determine properties of the functions p and q satisfying some special systems of differential inequalities, and extend their results to differential inclusions and subordinations.

On Duality in Second-Order Discrete and Differential Inclusions with Delay

The present paper studies the duality theory for the Mayer problem with second-order evolution differential inclusions with delay and state constraints. Although all the proofs in the paper relating to dual problems are carried out in the case of delay, these results are new for problems without delay, too. To this end, first we use an auxiliary problem with second-order discrete- and discrete-approximate inclusions. Second, applying infimal convolution concept of convex functions, step-by-step, we construct the dual problems for discrete, discrete–approximate, and differential inclusions, and prove duality results, where the Euler–Lagrange type inclusions are “duality relations” for both primary and dual problems and that the dual problem for discrete–approximate problem make a bridge between the dual problems of discrete and continuous problems. Thus, proceeding to the limit procedure, we establish the dual problem for the continuous problem. In addition, semilinear problems with discrete and differential inclusions of second order with delay are also considered. These problems show that supremum in the dual problems are realized over the set of solutions of the Euler–Lagrange-type discrete/differential inclusions, respectively.

On duality in optimal control problems with second-order differential inclusions and initial-point constraints

On duality in optimal control problems with second-order differential inclusions and initial-point constraints

On the impulsive Dirichlet problem for second-order differential inclusions

On the impulsive Dirichlet problem for second-order differential inclusions

Existence results for second-order stochastic differential inclusions driven by Lévy noise

In this paper we prove the existence of mild solutions for a second-order impulsive semilinear stochastic differential inclusion with a standard cylindrical Wiener process and Poisson jumps. We consider the case in which the right hand side can be either convex-valued.

Coupled System of Second-Order Stochastic Neutral Differential Inclusions Driven by Wiener Process and Poisson Jumps

In this paper we prove the existence of mild solutions for a second-order impulsive semilinear stochastic differential inclusion with an infinite-dimensional standard cylindrical Wiener process and Poisson jumps. We consider non convex-valued cases.