Applied Mathematics and Optimization最新文献

We study the exact controllability problem for the wave equation on a general finite metric graph with the Kirchhoff–Neumann matching conditions. Among all vertices and edges we choose certain active vertices and edges, and give a constructive proof that the wave equation on the graph is exactly controllable if (H^1(0,T)') Neumann controllers are placed at the active vertices and (L^2(0,T)) Dirichlet controllers are placed at the active edges. For such controls, we describe the state spaces for which our initial boundary value problem is well posed. The proofs for the shape and velocity controllability are purely dynamical, while the proof for the exact controllability utilizes both dynamical and spectral (moment method) approaches. The control time for this construction is determined by the chosen orientation and path decomposition of the graph.

In this work, we study a wave equation with nonlocal boundary damping of energy type. We begin by establishing the well-posedness of the problem using the Galerkin method. Next, we investigate the asymptotic behavior of the solution by applying the multiplier method, and we enhance the decay rate through the use of Nakao’s Lemma. Finally, we employ the radial multiplier technique to obtain an optimal polynomial decay rate under this type of damping.

In this work we introduce the concept of generalized exponential ({mathfrak {D}}_{mathcal {C}^*})–pullback attractors for evolution processes, which are compact and positively invariant families that pullback attract all elements of a universe of families ({mathfrak {D}}_{mathcal {C}^*}), with an exponential rate. Such concept, within the pullback framework for nonautonomous problems, was introduced in Bortolan et al. (Appl Math Optim 89(62):1–52, 2024) for more general decay functions (which include the exponential decay), but for fixed bounded sets rather than for a universe of families, and was inspired by Zhao et al. (Estimate of the attractive velocity of attractors for some dynamical systems, http://arxiv.org/abs/2108.07410, 2021), which dealt with the autonomous case. We prove a result that ensures the existence of a generalized exponential ({mathfrak {D}}_{mathcal {C}^*})–pullback attractor for an evolution process, using the concept of pullback (kappa )–dissipativity for evolution processes with respect to a general universe ({mathfrak {D}}). This required an adaptation of the results presented in Bortolan et al. (Appl Math Optim 89(62):1–52, 2024), which only covered the case of a polynomial rate of attraction for fixed bounded sets. Later, we prove that a nonautonomous wave equation has a generalized exponential ({mathfrak {D}}_{mathcal {C}^*})–pullback attractor. This, in turn, also implies the existence of the ({mathfrak {D}}_{mathcal {C}^*})–pullback attractor for such problem.

We establish the global-in-time well-posedness for a broad class of mean field games including those with the small mean field sensitivity and the linear-quadratic setting as special cases. Instead of using the master equation approach, we adopt the maximum principle to investigate the unique existence of the equilibrium strategy by solving the corresponding forward-backward stochastic differential equations (FBSDEs), whose global existence is shown by controlling the sensitivity of the backward solutions with respect to the initial data via new a priori estimates for the corresponding Jacobian flows. Besides, we provide the state-of-the-art study with general cost functions having both quadratic growth and non-convexity in the state variable. We also impose the structural conditions on the cost functions but not on the Hamiltonian. The advantages of this framework are threefold: (i) the structural conditions can be easily verified; (ii) reduced regularity of cost functions suffices for the unique existence of equilibrium solutions compared to solving the master equations; and (iii) when the mean field effect is not small, the cost functions are not convex in the state variable, or there is lack of monotonicity of cost functions, an accurate lifespan for the local existence of the FBSDEs is still given, which is not small in general. Finally, we provide a counterexample to illustrate the ill-posedness of the mean field games when the small mean field effect and the contemporary monotonicity conditions are violated, this demonstrates numerically that our assumptions should be sharp.

We propose several modifications of the Barzilai–Borwein (BB) step size in the variance reduction (VR) methods for finite-sum optimization problems. Our first approach relies on a scalar function, which we call the TaiL Function (TLF). The TLF maps the computed BB step size to some positive real number, which will be used as the step size instead. The computational overhead is almost negligible and the functional forms of TLFs in this work don’t involve any problem-dependent parameters. In the strongly convex setting, due to the undesirable appearance of the condition number (kappa ) in the linear convergence rate, the IFO complexity of VR methods with BB step size has the form (mathcal {O}((n+kappa ^a)kappa log (1/epsilon ))), (ain mathbb {R}_{+}). With the utilization of the TLF, the aforementioned complexity is improved to (mathcal {O}((n+kappa ^{tilde{a}})log (1/epsilon ))), (tilde{a}in mathbb {R}_{+}, tilde{a}<a). In the non-convex setting, we improve (mathcal {O}(n+nepsilon ^{-1})) of SVRG-SBB to (mathcal {O}(n+n^{beta }epsilon ^{-1})), where (beta in mathbb {R}_{+}) can take any value in (2/3, 1). Specifically, the constant step size regime is recovered by taking the TLF as a constant function, whose function value relies on problem-dependent parameters. As a counterpart of the constant step size regime, we also propose a BB-based vibration technique to set step sizes for VR methods, leading to methods with novel one-parameter step sizes. These methods have the same complexities compared to their constant step size versions. Meanwhile, they are more robust w.r.t. the sole step size parameter empirically. Moreover, a novel analysis is proposed for SARAH-I-type methods in the strongly convex setting. Numerical tests corroborate the proposed methods.

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.