Applied Mathematics and Optimization最新文献

In this paper, we study the social optima for a kind of time-inconsistent linear-quadratic (LQ) mean-field problem. We first propose the concept of (varepsilon)-social equilibrium strategy for our problem within the game theoretic framework. Next, an auxiliary time-inconsistent control problem is constructed by the scheme of person-by-person optimality and a set of linear feedback equilibrium controls for this auxiliary problem is presented explicitly via several ordinary differential equations (ODEs). Then we can derive a new kind of consistency condition (CC) system consisting of a forward stochastic differential equation (SDE) and three flows of backward stochastic differential equations (BSDEs), and obtain a set of decentralized admissible controls for the original problem. The wellposedness of this kind of CC system is also discussed in this paper. We prove that the set of admissible controls given above is indeed an (varepsilon)-social equilibrium strategy for our original problem. Finally, a numerical example is presented to illustrate our theoretic results visually.

This paper deals with the homogenization of a binary three-dimensional structure made of damped Helmholtz resonators embedded in a rigid porous matrix when both parts of each resonator, namely the duct and the chamber, are filled with the same Stokes fluid and when the Darcy and Stokes flows are coupled by a Beavers–Joseph type condition. As the small period of the distribution shrinks to zero, we study the asymptotic behaviour of the flow when the permeability and transfer coefficients on the interface are of unity order in contrast with the periodic distribution of resonators characterized by highly conductive parallel thin ducts. To that aim the energetic procedure of homogenization is associated to the control-zone method taking into account the geometry of the microstructure and heterogeneities of the conductivity coefficients. The main result lies in the asymptotic behaviour of the periodic distribution in the binary medium characterized by a significant increase of conductivity in thin ducts whose everlasting action contrasts with their local vanishing volume. The dependence with respect to the relative thickness of the ducts is studied.

The chemotaxis model

is considered under the boundary conditions (frac{partial u}{partial nu }-chi g(u)frac{partial v}{partial nu }=0) and (v=v_*) on (partial Omega), where (Omega subset {mathbb {R}}^n) ((nin {2,3})) is a ball and (v_*) is a given positive constant. Here, the parameters (chi ,k,mu) are positive and the function (gin C^1([0,infty ))) satisfies (0le g(u)le u^{beta }) with (frac{5}{6}le beta <1). For all suitably regular initial data, the present work provides a result on global boundedness of the radially symmetric classical solution in two dimensions when (beta =chi) and (1<l<frac{5}{3}), while the global existence of the radially symmetric weak solution is established in three-dimensional settings.

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.