Applied Mathematics and Optimization最新文献

In this work, we study the exponential stability of a piezoelectric beam subjected to magnetic effects and dissipation via microtemperatures. The model is formulated as a system of coupled partial differential equations, including a thermal equation to capture the dissipative effects at a microscopic scale. First, we establish the well-posedness of the problem within the framework of semigroup theory. Then, we use the energy method to derive sufficient conditions for the exponential stability of the system. Additionally, we conduct numerical experiments using finite differences to validate the theoretical results. Our findings indicate that the inclusion of dissipation due to microtemperatures plays a significant role in stabilizing the system, reducing undesired oscillations, and improving the energy decay rate.

In this paper we will provide conditions to explicitly calculate fractional powers and semigroup generation of (2 times 2) upper triangular matrices. Once this is done, we apply a Schur decomposition technique to (2times 2) matrix operators in order to reduce it to upper triangular and use the previous abstract theory to obtain explicit formulas for its fractional power and the semigroup it generates. This technique on Schur decomposition will be applied at two well-known examples from the context of partial differential equations: the Fitzhugh–Nagumo equation and the strongly damped wave equation. In particular, we will be able to provide the explicit formulation for the fractional version of those problems as well as their explicit solutions.

This paper studies robust time-inconsistent (TIC) linear-quadratic stochastic control problems, formulated by stochastic differential games. By a spike variation approach, we derive sufficient conditions for achieving the Nash equilibrium, which corresponds to a time-consistent (TC) robust policy, under mild technical assumptions. To illustrate our framework, we consider two scenarios of robust mean-variance analysis, namely with state- and control-dependent ambiguity aversion. We find numerically that with time inconsistency haunting the dynamic optimal controls, the ambiguity aversion enhances the effective risk aversion faster than the linear, implying that the ambiguity in the TIC cases is more impactful than that under the TC counterparts, e.g., expected utility maximization problems.

In this paper, the limiting behavior of invariant measures is mainly investigated for a class of stochastic quasilinear parabolic equations with nonlinear noise on thin domains. The existence and uniqueness of invariant measure on ((n+1))-dimensional thin domains are presented. The difficulty on estimates of the solutions for such problems in Sobolev space in the sense of thin domains is overcome by a novel proof techniques. Hence, the research results reveal that any limit of invariant measures of original equations on thin domains must be an invariant measure of the limiting equations when the ((n+1))-dimensional thin domains degenerates onto the n-dimensional space.

In this paper we study a bilinear optimal control problem related to a 2D chemo-repulsion stationary model with nonlinear production term. Firstly, we prove the existence of strong solutions for each control given; then, for the extremal problem, we prove the existence of at least one global optimal solution. Afterwards, using a generic result on the existence of Lagrange multipliers, we obtain the so-called first-order necessary optimality conditions for local optimal solutions. Furthermore, we discuss an extension of the results to 3D domains.

In this paper, we deal with the following nonlocal double phase problem with general growth conditions

where (alpha ,beta in (0,1)), (1<qle p<N/alpha ), (lambda in mathbb {R}), ((-Delta )_{p,a}^alpha +(-Delta )^beta _q) is the fractional (p, q)-Laplacian with weight ({a:mathbb {R}^Ntimes mathbb {R}^N}rightarrow mathbb {R}^+), (q<r<p+frac{alpha pq}{N}), (varepsilon >0) and (bin L^infty (mathbb {R}^N), hin C(mathbb {R})). Such equations can be used to model anisotropic materials in which the geometric shape of composite materials made of two different materials is determined by the function a. Since the nonlinear term h may satisfy Sobolev critical or supercritical growth, we first consider a truncated problem and study the existence of normalized solutions by combining the fractional Gagliardo-Nirenberg inequality with variational methods. We show that any normalized solution of the truncated problem is also a solution of our problem. This is achieved by estimating the bound of solutions using the De Giorgi iteration technique. Then we reveal that the multiplicity of normalized ground state solutions may be caused by the geometric shape of composite materials. More precisely, we prove that the number of normalized ground state solutions is at least the number of intersections between the minimum points of function a and the maximum points of function b as (varepsilon ) is small enough. Moreover, we discuss the asymptotic behavior of normalized solutions as (varepsilon rightarrow 0^+). Finally, the orbital stability of the ground state set of the problem is investigated. The main features of this paper are that the operator ((-Delta )_{p,a}^alpha +(-Delta )^beta _{q}) may generate double phase energy, and that the nonlinear term h may have Sobolev critical or supercritical growth at infinity. Our results are new even in the (p, q)-Laplacian case, i.e. when (alpha =beta =1).

In this paper, we develop an inexact version of the catching-up algorithm for sweeping processes. We define a new notion of approximate projection, which is compatible with any numerical method for approximating exact projections, as this new notion is not restricted to remain strictly within the set. We provide several properties of the new approximate projections, which enable us to prove the convergence of the inexact catching-up algorithm in three general frameworks: prox-regular moving sets, subsmooth moving sets, and merely closed sets. Additionally, we apply our numerical results to address complementarity dynamical systems, particularly electrical circuits with ideal diodes. In this context, we implement the inexact catching-up algorithm using a primal-dual optimization method, which typically does not necessarily guarantee a feasible point. Our results are illustrated through an electrical circuit with ideal diodes. Our results recover classical existence results in the literature and provide new insights into the numerical simulation of sweeping processes.

We study an approximate controllability problem for the continuity equation and its application to constructing transport maps with normalizing flows. Specifically, we construct time-dependent controls (theta =(w, a, b)) in the vector field (xmapsto w(a^top x + b)_+) to approximately transport a known base density (rho _{textrm{B}}) to a target density (rho _*). The approximation error is measured in relative entropy, and (theta ) are constructed piecewise constant, with bounds on the number of switches being provided. Our main result relies on an assumption on the relative tail decay of (rho _*) and (rho _{textrm{B}}), and provides hints on characterizing the reachable space of the continuity equation in relative entropy.

In this paper, we consider the inviscid limit problem to the higher dimensional incompressible Navier–Stokes equations in the whole space. It was proved in [Guo et al. J. Funct. Anal. 276:2821–2830, 2019] that given initial data (u_0in B^{s}_{p,r}) with (1le r<infty), the solution of the Navier–Stokes equations converges strongly in (B^{s}_{p,r}) to the solution of the Euler equations as the viscosity parameter tends to zero. In the case when (r=infty), we prove the failure of the (B^{s}_{p,infty })-convergence of the Navier-Stokes equations toward the Euler equations in the inviscid limit.

In this paper, we study finite-agent linear-quadratic games on graphs. Specifically, we propose a comprehensive framework that extends the existing literature by incorporating heterogeneous and interpretable player interactions. Compared to previous works, our model offers a more realistic depiction of strategic decision-making processes. For general graphs, we establish the convergence of fictitious play, a widely-used iterative solution method for determining the Nash equilibrium of our proposed game model. Notably, under appropriate conditions, this convergence holds true irrespective of the number of players involved. For vertex-transitive graphs, we develop a semi-explicit characterization of the Nash equilibrium. Through rigorous analysis, we demonstrate the well-posedness of this characterization under certain conditions. We present numerical experiments that validate our theoretical results and provide insights into the intricate relationship between various game dynamics and the underlying graph structure.