Continuous Dependence Theorems Research Articles

Continuous dependence in thermopiezoelectricity of nonsimple materials

AbstractIn this paper, we consider the linear theory for a model of a thermopiezoelectric nonsimple body as presented in Passarella and Tibullo (Entropy, https://doi.org/10.3390/e24091229. 2022) in which the second displacement gradient and the second electric potential gradient are included in the set of independent constitutive variables, adopting an entropy production inequality model proposed by Green and Laws. We set a mixed initial-boundary value problem and starting from two solutions we establish a reciprocity relation for the general case, which extends the analogue in Ciarletta et al. (Mech Res Commun, https://doi.org/10.1016/j.mechrescom.2022.103969, 2022), and two continuous dependence theorems for centrosymmetric materials. Both theorems consider the difference of two solutions of the problem: the first one depends only from supply terms, which are the external body forces, electric charge density and heat supply and the second one only depends from initial data for displacement, velocity, relative temperature and entropy fields. All these data, together with the boundary conditions, are assigned.

Input-to-state stability of positive delayed neural networks via impulsive control

This paper is concerned with the positivity and impulsive stabilization of equilibrium points of delayed neural networks (DNNs) subject to bounded disturbances. With the aid of the continuous dependence theorem for impulsive delay differential equations, a relaxed positivity condition is derived, which allows the neuron interconnection matrix to be Metzler if the activation functions satisfy a certain condition. The notion of input-to-state stability (ISS) is introduced to characterize internal global stability and disturbance attenuation performance for impulsively controlled DNNs. The ISS property is analyzed by employing a time-dependent max-separable Lyapunov function which is able to capture the positivity characterization and hybrid structure of the considered DNNs. A ranged dwell-time-dependent ISS condition is obtained, which allows to design an impulsive control law via partial state variables. As a byproduct, an improved global exponential stability criterion for impulse-free positive DNNs is obtained. The applicability of the achieved results is illustrated through three numerical examples.

Stochastic differential equations in infinite dimensional Hilbert space and its optimal control problem with Lévy processes

<abstract><p>The paper is concerned with a class of stochastic differential equations in infinite dimensional Hilbert space with random coefficients driven by Teugels martingales which are more general processes and the corresponding optimal control problems. Here Teugels martingales are a family of pairwise strongly orthonormal martingales associated with Lévy processes (see Nualart and Schoutens <sup>[<xref ref-type="bibr" rid="b21">21</xref>]</sup>). There are three major ingredients. The first is to prove the existence and uniqueness of the solutions by continuous dependence theorem of solutions combining with the parameter extension method. The second is to establish the stochastic maximum principle and verification theorem for our optimal control problem by the classic convex variation method and dual techniques. The third is to represent an example of a Cauchy problem for a controlled stochastic partial differential equation driven by Teugels martingales which our theoretical results can solve.</p></abstract>

Uniqueness, continuous dependence and reciprocity theorems in thermoelastic relaxed micromorphic continuum

A mixed initial-boundary value problem is formulated for linear thermo-elastic relaxed micromorphic continuum with asymmetric micro-distortion tensor, micro-dislocation tensor and temperature field. First, Lagrange identity involving two admissible processes at different instants for the considered mixed initial-boundary value problem is established. Using this identity, four theorems are proved: (i) uniqueness of solution without recourse either to an energy conservation law or to any boundedness assumptions on the thermoelastic coefficients, (ii) continuous dependence of solution of the problem on loads, (iii) continuous dependence of solution of the problem on the initial data, and (iv) reciprocity relation. The use of positive definiteness conditions on the thermoelastic coefficients is also avoided.

Continuous dependence of uncertain fractional differential equations with Caputo’s derivative

Uncertain fractional differential equation driven by Liu process plays an important role in describing uncertain dynamic systems. This paper investigates the continuous dependence of solution on the parameters and initial values, respectively, for uncertain fractional differential equations involving the Caputo fractional derivative in measure sense. Several continuous dependence theorems are obtained based on uncertainty theory by employing the generalized Gronwall inequality, in which the coefficients of uncertain fractional differential equation are required to satisfy the Lipschitz conditions. Several illustrative examples are provided to verify the validity of the obtained results.

Solutions to fractional Sobolev-type integro-differential equations in Banach spaces with operator pairs and impulsive conditions

We are concerned with fractional Sobolev-type integro-differential equations in Banach spaces with operator pairs and impulsive conditions, where the operator pairs generate propagation families. With the help of the theory of propagation family and Laplace transforms, along with an estimate for a special sequence improved in this article, we introduce a definition of mild solutions to the impulsive problem for these abstract fractional Sobolev-type integro-differential equations and we establish general existence theorems and a continuous dependence theorem, which essentially extend some previous conclusions. In our results, the operator B could be unbounded, and the existence of an operator B−1 is not necessarily needed. Moreover, we give some examples to illustrate our main results.

Forward and Backward Mean-Field Stochastic Partial Differential Equation and Optimal Control

This paper is mainly concerned with the solutions to both forward and backward mean-field stochastic partial differential equation and the corresponding optimal control problem for mean-field stochastic partial differential equation. The authors first prove the continuous dependence theorems of forward and backward mean-field stochastic partial differential equations and show the existence and uniqueness of solutions to them. Then they establish necessary and sufficient optimality conditions of the control problem in the form of Pontryagin’s maximum principles. To illustrate the theoretical results, the authors apply stochastic maximum principles to study the infinite-dimensional linear-quadratic control problem of mean-field type. Further, an application to a Cauchy problem for a controlled stochastic linear PDE of mean-field type is studied.

Stochastic evolution equations of jump type with random coefficients: existence, uniqueness and optimal control

We study a class of stochastic evolution equations of jump type with random coefficients and its optimal control problem. There are three major ingredients. The first is to prove the existence and uniqueness of the solutions by continuous dependence theorem of solutions combining with the parameter extension method. The second is to establish the stochastic maximum principle and verification theorem for our optimal control problem by the classic convex variation method and dual technique. The third is to represent an example of a Cauchy problem for a controlled stochastic partial differential equation with jumps which our theoretical results can solve.

Averaging of fuzzy integral equations

The integral equations are encountered in various fields of science and in numerous applications, including elasticity, plasticity, heat and mass transfer, oscillation theory, fluid dynamics, filtration theory, electrostatics, electrodynamics, biomechanics, game theory, control, queuing theory, electrical engineering, economics, and medicine. In this paper the fuzzy integral equation is considered and the existence and uniqueness theorem, the theorem of continuous dependence on the right-hand side and initial fuzzy set are proved. Also the possibility of using the scheme of full averaging for fuzzy integral equation with a small parameter is considered.

Some Theorems in the Theory of Microstretch Thermomagnetoelectroelasticity

The linear dynamic theory of microstretch thermomagnetoelectroelasticity is studied in this paper. First, a reciprocity relation which involves two processes at different instants is established to form the basis of a uniqueness result and a reciprocal theorem. The proof of the reciprocal theorem avoids both using the Laplace transform and incorporating the initial conditions into the equations of motion. The uniqueness theorem is derived with no definiteness assumption on the elastic constitutive coefficients. Then the continuous dependence theorem is discussed upon two external data systems. Finally, the variational principle of Hamilton type which fully characterizes the solution of the mixed boundary-initial-value problem (mixed problem) is obtained. These theorems lay a solid foundation for further theoretical and numerical studies on microstretch thermomagnetoelectroelastic materials.

On the Continuity in a Parameter for the Solutions of Boundary-Value Problems Total with Respect to the Spaces C (n+r)[a, b

We study a broad class of linear boundary-value problems for systems of ordinary differential equations, namely, the problems total with respect to the space C (n+r)[a, b], where n ∈ ℕ and r is the order of the equations. For their solutions, we prove the theorem of existence, uniqueness, and continuous dependence on the parameter in this space.

Optimal Control for Stochastic Delay Evolution Equations

In this paper, we investigate a class of infinite-dimensional optimal control problems, where the state equation is given by a stochastic delay evolution equation with random coefficients, and the corresponding adjoint equation is given by an anticipated backward stochastic evolution equation. We first prove the continuous dependence theorems for stochastic delay evolution equations and anticipated backward stochastic evolution equations, and show the existence and uniqueness of solutions to anticipated backward stochastic evolution equations. Then we establish necessary and sufficient conditions for optimality of the control problem in the form of Pontryagin's maximum principles. To illustrate the theoretical results, we apply stochastic maximum principles to study two examples, an infinite-dimensional linear-quadratic control problem with delay and an optimal control of a Dirichlet problem for a stochastic partial differential equation with delay. Further applications of the two examples to a Cauchy problem for a controlled linear stochastic partial differential equation and an optimal harvesting problem are also considered.

Some properties of finite-time stable stochastic nonlinear systems

In this paper we study some properties of finite-time stable stochastic nonlinear systems. We begin by showing several continuous dependence theorems of solutions on initial values under some conditions on the coefficients of stochastic systems. We then derive some regular properties of its stochastic settling time for a finite-time stable stochastic nonlinear system. We show continuity, positive definiteness and boundedness of the expected stochastic settling time under appropriate conditions. Finally, a Lyapunov function is constructed by making use of the expectation of the stochastic settling time, and the infinitesimal generator of the stochastic system defined on the Lyapunov function is also given, and hence resulting in a converse Lyapunov theorem of finite-time stochastic stability.

Young measure solutions for a class of forward-backward convection-diffusion equations

This paper is devoted to the first initial boundary value problems of a class of forward-backward convection-diffusion equations. The existence theorem and the continuous dependence theorem of Young measure solutions are established.

Uniqueness and stability results for non-linear Johnson-Segalman viscoelasticity and related models

In this paper we have proved exponential asymptotic stability for the corotational incompressible diffusive Johnson-Segalman viscolelastic model and a simple decay result for the corotational incompressible hyperbolic Maxwell model. Moreover we have established continuous dependence and uniqueness results for the non-zero equilibrium solution.  &nbsp In the compressible case, we have proved a Hölder continuous dependence theorem upon the initial data and body force for both models, whence follows a result of continuous dependence on the initial data and, therefore, uniqueness.  &nbsp For the Johnson-Segalman model we have also dealt with the case of negative elastic viscosities, corresponding to retardation effects.A comparison with other type of viscoelasticity, showing short memory elastic effects, is given.

Continuous dependence theorems on solutions of uncertain differential equations

In ordinary differential equation (ODE) and stochastic differential equation (SDE), the solution continuously depends on initial value and parameter under some conditions. This paper investigates the analogous continuous dependence theorems in uncertain differential equation (UDE). It proves two continuous dependence theorems, a basic one and a general one.

Nonsimple material problems addressed by the Lagrange’s identity

Our paper is concerned with some basic theorems for nonsimple thermoelastic materials. By using the Lagrange identity, we prove the uniqueness theorem and some continuous dependence theorems without recourse to any energy conservation law, or to any boundedness assumptions on the thermoelastic coefficients. Moreover, we avoid the use of positive definiteness assumptions on the thermoelastic coefficients.

Backward doubly stochastic differential equations with infinite time horizon

We give a sufficient condition on the coefficients of a class of infinite horizon backward doubly stochastic differential equations (BDSDES), under which the infinite horizon BDSDES have a unique solution for any given square integrable terminal values. We also show continuous dependence theorem and convergence theorem for this kind of equations.

Stochastic PDEs and Infinite Horizon Backward Doubly Stochastic Differential Equations

We give a sufficient condition on the coefficients of a class of infinite horizon BDSDEs, under which the infinite horizon BDSDEs have a unique solution for any given square integrable terminal values. We also show continuous dependence theorem and convergence theorem for this kind of equations. A probabilistic interpretation for solutions to a class of stochastic partial differential equations is given.

Several Results on Uniqueness and Continuous Data Dependence in Thermoelasticity of Type III

In the present paper we consider the linear theory of thermoelasticity of type III for anisotropic and inhomogeneous materials as developed by Green and Naghdi (1992, 1995). We consider the general initial-boundary value problems associated with the theory in concern and then we identify some mild constitutive hypotheses that lead to appropriate energetic measures of solutions. On this basis we are able to prove some new uniqueness and continuous data dependence theorems. When the boundary surface is maintained at zero thermal displacement, we are able to obtain uniqueness and continuous data dependence results by using only the positiveness of the density mass, the positive definiteness of the elasticity tensor and the positive definiteness of the heat conductivity tensor, without any sign defined assumption upon the specific heat. These results are established by means of new techniques which are essentially different from those previously used for the classical theory of thermoelasticity.