Authors:Nikolaos S. Papageorgiou; Vicenţiu D. Rădulescu; Dušan D. Repovš Pages: 1 - 23 Abstract: We consider a nonlinear Neumann elliptic inclusion with a source (reaction term) consisting of a convex subdifferential plus a multivalued term depending on the gradient. The convex subdifferential incorporates in our framework problems with unilateral constraints (variational inequalities). Using topological methods and the Moreau-Yosida approximations of the subdifferential term, we establish the existence of a smooth solution. PubDate: 2018-08-01 DOI: 10.1007/s00245-016-9392-y Issue No:Vol. 78, No. 1 (2018)

Authors:Harold A. Moreno-Franco Pages: 25 - 60 Abstract: The main goal of this paper is to establish existence, regularity and uniqueness results for the solution of a Hamilton–Jacobi–Bellman (HJB) equation, whose operator is an elliptic integro-differential operator. The HJB equation studied in this work arises in singular stochastic control problems where the state process is a controlled d-dimensional Lévy process. PubDate: 2018-08-01 DOI: 10.1007/s00245-016-9397-6 Issue No:Vol. 78, No. 1 (2018)

Authors:Yuxiang Liu; Peng-Fei Yao Pages: 61 - 101 Abstract: Decay of the energy for the Cauchy problem of the wave equation on Riemannian manifolds with a variable damping term \(V(x)u_{t}\) is considered, where \(V(x) \ge V_{0}(1 + \rho ^{2})^{-\frac{1}{2}}\) ( \(\rho \) being a distance function under the Riemannian metric). Some relations among the decay rates of energy, the size of the coefficients \(V_{0}\) , and the radial curvatures of the Riemannian metric are presented. PubDate: 2018-08-01 DOI: 10.1007/s00245-017-9399-z Issue No:Vol. 78, No. 1 (2018)

Authors:J. Doll; P. Dupuis; P. Nyquist Pages: 103 - 144 Abstract: Parallel tempering, or replica exchange, is a popular method for simulating complex systems. The idea is to run parallel simulations at different temperatures, and at a given swap rate exchange configurations between the parallel simulations. From the perspective of large deviations it is optimal to let the swap rate tend to infinity and it is possible to construct a corresponding simulation scheme, known as infinite swapping. In this paper we propose a novel use of large deviations for empirical measures for a more detailed analysis of the infinite swapping limit in the setting of continuous time jump Markov processes. Using the large deviations rate function and associated stochastic control problems we consider a diagnostic based on temperature assignments, which can be easily computed during a simulation. We show that the convergence of this diagnostic to its a priori known limit is a necessary condition for the convergence of infinite swapping. The rate function is also used to investigate the impact of asymmetries in the underlying potential landscape, and where in the state space poor sampling is most likely to occur. PubDate: 2018-08-01 DOI: 10.1007/s00245-017-9401-9 Issue No:Vol. 78, No. 1 (2018)

Authors:Jingrui Sun; Jiongmin Yong Pages: 145 - 183 Abstract: This paper is concerned with stochastic linear quadratic (LQ, for short) optimal control problems in an infinite horizon with constant coefficients. It is proved that the non-emptiness of the admissible control set for all initial state is equivaleznt to the \(L^{2}\) -stabilizability of the control system, which in turn is equivalent to the existence of a positive solution to an algebraic Riccati equation (ARE, for short). Different from the finite horizon case, it is shown that both the open-loop and closed-loop solvabilities of the LQ problem are equivalent to the existence of a static stabilizing solution to the associated generalized ARE. Moreover, any open-loop optimal control admits a closed-loop representation. Finally, the one-dimensional case is worked out completely to illustrate the developed theory. PubDate: 2018-08-01 DOI: 10.1007/s00245-017-9402-8 Issue No:Vol. 78, No. 1 (2018)

Authors:Giuseppe Buttazzo; Thierry Champion; Luigi De Pascale Pages: 185 - 200 Abstract: We consider some repulsive multimarginal optimal transportation problems which include, as a particular case, the Coulomb cost. We prove a regularity property of the minimizers (optimal transportation plan) from which we deduce existence and some basic regularity of a maximizer for the dual problem (Kantorovich potential). This is then applied to obtain some estimates of the cost and to the study of continuity properties. PubDate: 2018-08-01 DOI: 10.1007/s00245-017-9403-7 Issue No:Vol. 78, No. 1 (2018)

Authors:Weiwei Hu Pages: 201 - 217 Abstract: We discuss the optimal boundary control problem for mixing an inhomogeneous distribution of a passive scalar field in an unsteady Stokes flow. The problem is motivated by mixing the fluids within a cavity or vessel at low Reynolds numbers by moving the walls or stirring at the boundary. It is natural to consider the velocity field which is induced by a control input tangentially acting on the boundary of the domain through the Navier slip boundary conditions. Our main objective is to design an optimal Navier slip boundary control that optimizes mixing at a given final time. This essentially leads to a finite time optimal control problem of a bilinear system. In the current work, we consider a general open bounded and connected domain \(\Omega \subset \mathbb {R}^{d}, d=2,3\) . We employ the Sobolev norm for the dual space \((H^{1}(\Omega ))'\) of \(H^{1}( \Omega )\) to quantify mixing of the scalar field in terms of the property of weak convergence. A rigorous proof of the existence of an optimal control is presented and the first-order necessary conditions for optimality are derived. PubDate: 2018-08-01 DOI: 10.1007/s00245-017-9404-6 Issue No:Vol. 78, No. 1 (2018)

Authors:Francesca Biagini; Alessandro Gnoatto; Maximilian Härtel Pages: 405 - 441 Abstract: We develop the HJM framework for forward rates driven by affine processes on the state space of symmetric positive semidefinite matrices. In this setting we find an explicit representation for the long-term yield in terms of the model parameters. This generalises the results of El Karoui et al. (Rev Deriv Res 1(4):351–369, 1997) and Biagini and Härtel (Int J Theor Appl Financ 17(3):1–24, 2012), where the long-term yield is investigated under no-arbitrage assumptions in a HJM setting using Brownian motions and Lévy processes respectively. PubDate: 2018-06-01 DOI: 10.1007/s00245-016-9379-8 Issue No:Vol. 77, No. 3 (2018)

Authors:P. Tamilalagan; P. Balasubramaniam Pages: 443 - 462 Abstract: In this manuscript, we investigate the solvability and optimal controls for fractional stochastic differential equations driven by Poisson jumps in Hilbert space by using analytic resolvent operators. Sufficient conditions are derived to prove that the system has a unique mild solution by using the classical Banach contraction mapping principle. Then, the existence of optimal control for the corresponding Lagrange optimal control problem is investigated. Finally, the derived theoretical result is validated by an illustrative example. PubDate: 2018-06-01 DOI: 10.1007/s00245-016-9380-2 Issue No:Vol. 77, No. 3 (2018)

Authors:R. Ferretti; A. Sassi; H. Zidani Abstract: Hybrid control systems are dynamical systems that can be controlled by a combination of both continuous and discrete actions. In this paper we study the approximation of optimal control problems associated to this kind of systems, and in particular of the quasi-variational inequality which characterizes the value function. Our main result features the error estimates between the value function of the problem and its approximation. We also focus on the hypotheses describing the mathematical model and the properties defining the class of numerical scheme for which the result holds true. PubDate: 2018-08-09 DOI: 10.1007/s00245-018-9515-8

Authors:Marcelo M. Cavalcanti; Emanuela R. S. Coelho; Valéria N. Domingos Cavalcanti Abstract: This paper is concerned with the study of a transmission problem of viscoelastic waves with hereditary memory, establishing the existence, uniqueness and exponential stability for the solutions of this problem. The proof of the stabilization result combines energy estimates and results due to Gérard (Commun Partial Differ Equ 16:1761–1794 (1991)) on microlocal defect measures. PubDate: 2018-08-02 DOI: 10.1007/s00245-018-9514-9

Authors:Volodymyr Hrynkiv; Sergiy Koshkin Abstract: An optimal control of a steady state thermistor problem is considered where the convective boundary coefficient is taken to be the control variable. A distinct feature of this paper is that the problem is considered in \({\mathbb {R}}^d\) , where \(d>2\) , and the electrical conductivity is allowed to vanish above a threshold temperature value. Existence of the state system is proved. An objective functional is introduced, existence of the optimal control is proved, and the optimality system is derived. PubDate: 2018-08-01 DOI: 10.1007/s00245-018-9511-z

Authors:Diogo A. Gomes; João Saúde Abstract: Here, we develop numerical methods for finite-state mean-field games (MFGs) that satisfy a monotonicity condition. MFGs are determined by a system of differential equations with initial and terminal boundary conditions. These non-standard conditions make the numerical approximation of MFGs difficult. Using the monotonicity condition, we build a flow that is a contraction and whose fixed points solve both for stationary and time-dependent MFGs. We illustrate our methods with a MFG that models the paradigm-shift problem. PubDate: 2018-08-01 DOI: 10.1007/s00245-018-9510-0

Authors:Tianxiao Wang Abstract: This paper deals with a class of time inconsistent stochastic linear quadratic optimal control problems in Markovian framework. Three notions, i.e., closed-loop equilibrium strategies, open-loop equilibrium controls and open-loop equilibrium strategies, are characterized in unified manners. These results indicate clearer and deeper distinctions among these notions. For example, in particular time consistent setting, the open-loop equilibrium controls are fully characterized by first-order, second-order necessary optimality conditions, and are not optimal in general, while the closed-loop equilibrium controls naturally reduce into closed-loop optimal controls. PubDate: 2018-07-26 DOI: 10.1007/s00245-018-9513-x

Authors:Gabriela Marinoschi Abstract: We are concerned with a nonlinear nonautonomous model represented by an equation describing the dynamics of an age-structured population diffusing in a space habitat O, governed by local Lipschitz vital factors and by a stochastic behavior of the demographic rates possibly representing emigration, immigration and fortuitous mortality. The model is completed by a random initial condition, a flux type boundary conditions on \(\partial O\) with a random jump in the population density and a nonlocal nonlinear boundary condition given at age zero. The stochastic influence is expressed by a linear multiplicative Gaussian noise perturbation in the equation. The main result proves that the stochastic model is well-posed, the solution being in the class of path-wise continuous functions and satisfying some particular regularities with respect to the age and space. The approach is based on a rescaling transformation of the stochastic equation into a random deterministic time dependent hyperbolic-parabolic equation with local Lipschitz nonlinearities. The existence and uniqueness of a strong solution to the random deterministic equation is proved by combined semigroup, variational and approximation techniques. The information given by these results is transported back via the rescaling transformation towards the stochastic equation and enables the proof of its well-posedness. PubDate: 2018-07-25 DOI: 10.1007/s00245-018-9507-8

Authors:Igor G. Vladimirov; Ian R. Petersen; Matthew R. James Abstract: This paper is concerned with risk-sensitive performance analysis for linear quantum stochastic systems interacting with external bosonic fields. We consider a cost functional in the form of the exponential moment of the integral of a quadratic polynomial of the system variables over a bounded time interval. Such functionals are related to more conservative behaviour and robustness of systems with respect to statistical uncertainty, which makes the challenging problems of their computation and minimization practically important. To this end, we obtain an integro-differential equation for the time evolution of the quadratic–exponential functional, which is different from the original quantum risk-sensitive performance criterion employed previously for measurement-based quantum control and filtering problems. Using multi-point Gaussian quantum states for the past history of the system variables and their first four moments, we discuss a quartic approximation of the cost functional and its infinite-horizon asymptotic behaviour. The computation of the asymptotic growth rate of this approximation is reduced to solving two algebraic Lyapunov equations. Further approximations of the cost functional, based on higher-order cumulants and their growth rates, are applied to large deviations estimates in the form of upper bounds for tail distributions. We discuss an auxiliary classical Gaussian–Markov diffusion process in a complex Euclidean space which reproduces the quantum system variables at the level of covariances but has different fourth-order cumulants, thus showing that the risk-sensitive criteria are not reducible to quadratic–exponential moments of classical Gaussian processes. The results of the paper are illustrated by a numerical example and may find applications to coherent quantum risk-sensitive control problems, where the plant and controller form a fully quantum closed-loop system, and other settings with nonquadratic cost functionals. PubDate: 2018-07-24 DOI: 10.1007/s00245-018-9512-y

Authors:Ivan Yegorov; Peter M. Dower Abstract: This paper extends the considerations of the works (Darbon and Osher, Res Math Sci 3:19, 2016; Chow et al., 2017, arxiv.org/abs/1704.02524) regarding curse-of-dimensionality-free numerical approaches to solve certain types of Hamilton–Jacobi equations arising in optimal control problems, differential games and elsewhere. A rigorous formulation and justification for the extended Hopf–Lax formula of(Chow et al., 2017, arxiv.org/abs/1704.02524) is provided together with novel theoretical and practical discussions including useful recommendations. By using the method of characteristics, the solutions of some problem classes under convexity/concavity conditions on Hamiltonians (in particular, the solutions of Hamilton–Jacobi–Bellman equations in optimal control problems) are evaluated separately at different initial positions. This allows for the avoidance of the curse of dimensionality, as well as for choosing arbitrary computational regions. The corresponding feedback control strategies are obtained at selected positions without approximating the partial derivatives of the solutions. The results of numerical simulations demonstrate the high potential of the proposed techniques. It is also pointed out that, despite the indicated advantages, the related approaches still have a limited range of applicability, and their extensions to Hamilton–Jacobi–Isaacs equations in zero-sum two-player differential games are currently developed only for sufficiently narrow classes of control systems. These extensions require further investigation. PubDate: 2018-07-24 DOI: 10.1007/s00245-018-9509-6

Authors:Josef Janák Abstract: Stochastic partial differential equations of second order with two unknown parameters are studied. Based on ergodicity, two suitable families of minimum contrast estimators are introduced. Strong consistency and asymptotic normality of estimators are proved. The results are applied to hyperbolic equations perturbed by Brownian noise. PubDate: 2018-07-13 DOI: 10.1007/s00245-018-9506-9

Authors:Amir Peyravi Abstract: In this work we investigate asymptotic stability and instability at infinity of solutions to a logarithmic wave equation $$\begin{aligned} u_{tt}-\Delta u + u + (g\,*\, \Delta u)(t)+ h(u_{t})u_{t}+ u ^{2}u=u\log u ^{k}, \end{aligned}$$ in an open bounded domain \(\Omega \subseteq \mathbb {R}^3\) whith \(h(s)=k_{0}+k_{1} s ^{m-1}.\) We prove a general stability of solutions which improves and extends some previous studies such as the one by Hu et al. (Appl Math Optim, https://doi.org/10.1007/s00245-017-9423-3) in the case \(g=0\) and in presence of linear frictional damping \(u_{t}\) when the cubic term \( u ^2u\) is replaced with u. In the case \(k_{1}=0,\) we also prove that the solutions will grow up as an exponential function. Our result shows that the memory kernel g dose not need to satisfy some restrictive conditions to cause the unboundedness of solutions. PubDate: 2018-07-11 DOI: 10.1007/s00245-018-9508-7

Authors:Kamel Hamdache; Djamila Hamroun Abstract: We study a nonlinear coupling system of partial differential equations describing the dynamic of a magnetic fluid with internal rotations. The present mathematical model generalizes those discussed previously in the literature since actually the fluid is electrically conducting inducing additional nonlinearities in the problem and the dynamics of the magnetic field is described by the quasi-static Maxwell equations instead of the usual magnetostatic ones. We prove existence of weak solutions with finite energy first for the unsteady problem then for the steady one. PubDate: 2018-06-14 DOI: 10.1007/s00245-018-9505-x