Non-autonomous Problems Research Articles

On the existence and multiplicity of eigenvalues for a class of double-phase non-autonomous problems with variable exponent growth

On the existence and multiplicity of eigenvalues for a class of double-phase non-autonomous problems with variable exponent growth

Well-posedness for a general class of differential inclusions

We consider an abstract class of differential inclusions, which covers differential-algebraic and non-autonomous problems as well as problems with delay. Under weak assumptions on the operators involved, we prove the well-posedness of those differential inclusions in a pure Hilbert space setting. Moreover, we study the causality of the associated solution operator. The theory is illustrated by an application to a semistatic quasilinear variant of Maxwell's equations.

A Du Bois-Reymond Convex Inclusion for Nonautonomous Problems of the Calculus of Variations and Regularity of Minimizers

We consider a local minimizer, in the sense of the \(W^{1,m}\) norm (\(m\ge 1\)), of the classical problem of the calculus of variations $$\begin{aligned} {\left\{ \begin{array}{ll} {\mathrm{Minimize}}\quad &{}\displaystyle I(x):=\int _a^b\varLambda (t,x(t), x'(t))\,dt+\varPsi (x(a), x(b))\\ \text {subject to:} &{}x\in W^{1,m}([a,b];\mathbb {R}^n),\\ &{}x'(t)\in C\,\text { a.e., } \,x(t)\in \varSigma \quad \forall t\in [a,b].\\ \end{array}\right. } \end{aligned}$$ (P) where \(\varLambda :[a,b]\times \mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathbb {R}\cup \{+\infty \}\) is just Borel measurable, C is a cone, \(\varSigma \) is a nonempty subset of \(\mathbb {R}^n\) and \(\varPsi \) is an arbitrary possibly extended valued function. When \(\varLambda \) is real valued, we merely assume a local Lipschitz condition on \(\varLambda \) with respect to t, allowing \(\varLambda (t,x,\xi )\) to be discontinuous and nonconvex in x or \(\xi \). In the case of an extended valued Lagrangian, we impose the lower semicontinuity of \(\varLambda (\cdot ,x,\cdot )\), and a condition on the effective domain of \(\varLambda (t,x,\cdot )\). We use a recent variational Weierstrass type inequality to show that the minimizers satisfy a relaxation result and an Erdmann – Du Bois-Reymond convex inclusion which, remarkably, holds whenever \(\varLambda (x,\xi )\) is autonomous and just Borel. Under a growth condition, weaker than superlinearity, we infer the Lipschitz continuity of minimizers.

Minimal residual multistep methods for large stiff non-autonomous linear problems

The purpose of this work is to introduce a new idea of how to avoid the factorization of large matrices during the solution of stiff systems of ODEs. Starting from the general form of an explicit linear multistep method we suggest to adaptively choose its coefficients on each integration step in order to minimize the norm of the residual of an implicit BDF formula. Thereby we reduce the number of unknowns on each step from n to O(1), where n is the dimension of the ODE system. We call this type of methods Minimal Residual Multistep (MRMS) methods. In the case of linear non-autonomous problem, besides the evaluations of the right-hand side of ODE, the resulting numerical scheme additionally requires one solution of a linear least-squares problem with a thin matrix per step. We show that the order of the method and its zero-stability properties coincide with those of the used underlying BDF formula.For the simplest analog of the implicit Euler method the properties of linear stability are investigated. Though the classical absolute stability analysis is not fully relevant to the MRMS methods, it is shown that this one-step method is applicable in stiff case. In the numerical experiment section we consider the fixed-step integration of a two-dimensional non-autonomous heat equation using the MRMS methods and their classical BDF counterparts. The starting values are taken from a preset slowly-varying exact solution. The comparison showed that both methods give similar numerical solutions, but in the case of large systems the MRMS methods are faster, and their advantage considerably increases with the growth of dimension. Python code with the experimental code can be downloaded from the GitHub repository https://github.com/bfaleichik/mrms.

The Relaxation of Complementary Slackness Conditions as a Regularization Method for Optimal Control Problems

A new approach to the transformation of solutions of optimal control problems based on the special form of relaxation of complementary slackness conditions is presented. The proposed approach is tested on the Russian banking system model, which is derived as a solution of a linear nonautonomous optimization problem with mixed constraints. It is shown that the use of this method regularizes the model in a sense it becomes applicable for the forecasting of the main Russian banking indicators.

Nonautonomous Conley index theory the connecting homomorphism

Attractor-repeller decompositions of isolated invariant sets give rise to so-called connecting homomorphisms. These homomorphisms reveal information on the existence and strucuture of connecting trajectories of the underlying dynamical system. To give a meaningful generalization of this general principle to nonautonomous problems, the nonautonomous homology Conley index is expressed as a direct limit. Moreover, it is shown that a nontrivial connecting homomorphism implies, on the dynamical systems level, a sort of uniform connectedness of the attractor-repeller decomposition.

Development of Improved Scheme for Numerical Integration of Autonomous and Non-Autonomous Initial Value Problems

In this research study, an explicit and single stepnew third order improved scheme for numerical integration of autonomous and non-autonomous initial value problemsalso called Cauchy problems in ordinary differential equations has been developed. Linear stability analysis with corresponding stability region is drawn and error analysis has been provided to confirm third order accuracy of the improved scheme. Inclusion of a partial derivative with respect to the dependent variable with in two slopes of the improved scheme has improved its efficiency in terms of local and global truncation errors. Finally, numerical examples are provided to show performance of the improved scheme in comparison with other existing methods having same order of local accuracy. Throughout the study it has been observed from the results and comparison of the improved scheme that develop algorithm is superior to some existing numerical methods of same order in term of error analysis as well as accuracy viewpoint.The software MATLAB is used to justify the results and graphical representation of the improved scheme.

Nonautonomous Conley index theory. Continuation of Morse-decompositions

In previous works the author established a nonautonomous Conley index based on the interplay between a nonautonomous evolution operator and its skew-product formulation. In this paper, the treatment of attractor-repeller decomposition is refined. The more general concept of partially ordered Morse-decompositions is used. It is shown that, in the nonautonomous setting, these Morse-decompositions persist under small perturbations. Furthermore, a continuation property for these Morse decompositions is established. Roughly speaking, the index of every Morse-set and every connecting homomorphism continue as the nonautonomous problem, depending continuously on a parameter, changes.

Nonresonance conditions on the potential for a nonlinear nonautonomous Neumann problem

The aim of this paper is to establish the existence of the principal eigencurve of the p-Laplacian operator with the nonconstant weight subject to Neumann boundary conditions. We then study the nonresonce phenomena under the first eigenvalue and under the principal eigencurve, thus we obtain existence results for some nonautonomous Neumann elliptic problems involving the p-Laplacian operator.

Trajectory and global attractors for generalized processes

In this work the theory of generalized processes is used to describe the dynamics of a nonautonomous multivalued problem and, through this approach, some conditions for the existence of trajectory attractors are proved. By projecting the trajectory attractor on the phase space, the uniform attractor for the multivalued process associated to the problem is obtained and some conditions to guarantee the invariance of the uniform attractor are given. Furthermore, the existence of the uniform attractor for a class of \begin{document}$ p $\end{document} -Laplacian non-autonomous problems with dynamical boundary conditions is established.

From manifolds to Lagrangian coherent structures in galactic bar models

We study the dynamics near the unstable Lagrangian points in galactic bar models using dynamical system tools in order to determine the global morphology of a barred galaxy. We aim at the case of non-autonomous models, in particular with secular evolution, by allowing the bar pattern speed to decrease with time. We have extended the concept of manifolds widely used in the autonomous problem to the Lagrangian coherent structures (LCS), widely used in fluid dynamics, which behave similar to the invariant manifolds driving the motion. After adapting the LCS computation code to the galactic dynamics problem, we apply it to both the autonomous and non-autonomous problems, relating the results with the manifolds and identifying the objects that best describe the motion in the non-autonomous case. We see that the strainlines coincide with the first intersection of the stable manifold when applied to the autonomous case, while, when the secular model is used, the strainlines still show the regions of maximal repulsion associated to both the corresponding stable manifolds and regions with a steep change of energy. The global morphology of the galaxy predicted by the autonomous problem remains unchanged.

A selected pullback attractor

We introduce a new class of multivalued processes generated by generalized processes and a new concept of pullback attractor, named a selected pullback attractor, and we establish suitable conditions for the existence of such a selected pullback attractor. Finally, we give and application to nonautonomous problems with dynamic boundary conditions.

Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control

This paper focuses on the development of necessary optimality conditions for nonautonomous optimal control problems with nonsmooth mixed state and control constraints. In most of the existing results, the necessary optimality conditions for nonautonomous optimal control problems with mixed state and control constraints are derived under the Mangasarian-Fromovitz condition or even stronger. In this paper we derive the necessary optimality conditions for nonautonomous optimal control problems under constraint qualifications which are weaker than Mangasarian-Fromovitz condition. Moreover necessary optimality conditions with an Euler inclusion taking a bounded explicit multiplier form are derived for certain cases. Specifying these results to bilevel optimal control problems with finite-dimensional lower level we obtain necessary optimality conditions under weaker qualification conditions.

Global Continuation of Homoclinic Solutions

When extending bifurcation theory of dynamical systems to nonautonomous problems, it is a central observation that hyperbolic equilibria persist as bounded entire solutions under small temporally varying perturbations. In this paper, we abandon the smallness assumption and aim to investigate the global structure of the entity of all such bounded entire solutions in the situation of nonautonomous difference equations. Our tools are global implicit function theorems based on an ambient degree theory for Fredholm operators due to Fitzpatrick, Pejsachowicz and Rabier. For this we yet have to restrict to so-called homoclinic solutions, whose limit is 0 in both time directions.

Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay

In this work we prove the existence of solution for a p-Laplacian non-autonomous problem with dynamic boundary and infinite delay. We ensure the existence of pullback attractor for the multivalued process associated to the non-autonomous problem we are concerned.

Forest land value and rotation with an alternative land use

We solve Faustmann's problem when the land manager plans to switch from the current tree species to some alternative species or land use. Such situations occur when the value of the alternative increases relative to the value of the species currently in place. The paper characterizes the land value function and the optimum rotations, highlighting the differences between this non-autonomous problem and the traditional Faustmann problem. We show that, from one harvest to the next until the switch, rotations can be constant and equal to the Faustmann rotation, or increasingly higher than the Faustmann rotation, or decreasingly lower. In the last two situations, the higher the number of previous harvests of the currently planted species before the switch to the alternative use, the closer the last rotation is to the Faustmann rotation.

Pullback attractors for a class of non-autonomous thermoelastic plate systems

In this article we study the asymptotic behavior of solutions, in the sense of pullback attractors, of the evolution system $\begin{cases}u_{tt} +Δ^2 u+a(t)Δθ=f(t,u),&t>τ,\ x∈Ω,\\θ_t-κΔ θ-a(t)Δ u_t=0,&t>τ,\ x∈Ω,\end{cases}$ subject to boundary conditions $u=Δ u=θ=0,\ t>τ,\ x∈\partial Ω,$ where $Ω$ is a bounded domain in $\mathbb{R}^N$ with $N≥ 2$, which boundary $\partialΩ$ is assumed to be a $\mathcal{C}^4$-hypersurface, $κ>0$ is constant, $a$ is an Hölder continuous function and $f$ is a dissipative nonlinearity locally Lipschitz in the second variable. Using the theory of uniform sectorial operators, in the sense of P. Sobolevskiǐ ([23]), we give a partial description of the fractional power spaces scale for the thermoelastic plate operator and we show the local and global well-posedness of this non-autonomous problem. Furthermore we prove existence and uniform boundedness of pullback attractors.

Pullback Attractors for Multivalued Processes and Application to Nonautonomous Problems with Dynamic Boundary Conditions

In this work we prove, by using the Faedo-Galerkin method, the existence of at least one solution for a class of p-Laplacian nonautonomous problems with dynamic boundary conditions. We also extend to the multivalued context results on the existence of \(\mathcal {D}\)-pullback attractor - the pullback attractor for a determined basin of attraction \(\mathcal {D}\) - and we prove the existence of such attractors for the class of nonautonomous problems we are concerned about.

Existence of solutions of abstract non-autonomous second order integro-differential equations

In this paper the existence of solutions of a non-autonomous abstract integro-differential equation of second order is considered. Assuming the existence of an evolution operator corresponding to the associate abstract non-autonomous Cauchy problem of second order, we establish the existence of a resolvent operator for the homogeneous integro-differential equation and the existence of mild solutions to the inhomogeneous integro-differential equation. Furthermore, we study the existence of classical solutions of the integro-differential equation. Finally, we apply our results to the study of the existence of solutions of a non-autonomous wave equation.

The real Schur decomposition estimates Lyapunov characteristic exponents with multiplicity greater than one

Lyapunov characteristic exponents are indicators of the nature and of the stability properties of solutions of differential equations. The estimation of Lyapunov exponents of algebraic multiplicity greater than 1 is troublesome. In this work, we intuitively derive an interpretation of higher multiplicity Lyapunov exponents in forms that occur in simple linear time invariant problems of engineering relevance. We propose a method to determine them from the real Schur decomposition of the state transition matrix of the linear, nonautonomous problem associated with the fiducial trajectory. So far, no practical way has been found to formulate the method as an algorithm capable of mitigating over- or underflow in the numerical computation of the state transition matrix. However, this interesting approach in some practical cases is shown to provide quicker convergence than usual methods like the discrete QR and the continuous QR and Singular Value Decomposition (SVD)methods when Lyapunov exponents with multiplicity greater than one are present.