Authors:Vanja Nikolić; Barbara Kaltenbacher Pages: 261 - 301 Abstract: Abstract We are interested in shape sensitivity analysis for an optimization problem arising in medical applications of high intensity focused ultrasound. The goal is to find the optimal shape of a focusing acoustic lens so that the desired acoustic pressure at a kidney stone is achieved. Coupling of the silicone acoustic lens and nonlinearly acoustic fluid region is modeled by the Westervelt equation with nonlinear strong damping and piecewise constant coefficients. We follow the variational approach to calculating the shape derivative of the cost functional which does not require computing the shape derivative of the state variable; however assumptions of certain spatial regularity of the primal and the adjoint state are needed to obtain the derivative, in particular for its strong form according to the Delfour–Hadamard–Zolésio structure theorem. PubDate: 2017-10-01 DOI: 10.1007/s00245-016-9340-x Issue No:Vol. 76, No. 2 (2017)

Authors:Junjun Kang; Yanbin Tang Pages: 303 - 321 Abstract: Abstract In this paper, we consider an option pricing problem in a pure jump model where the process X(t) models the logarithm of the stock price. By the Schauder fixed point theorem, we show the existence and uniqueness of the solutions in H \(\ddot{o}\) lder spaces for the European and American option pricing problems respectively. Due to the estimates of fractional heat kernel, we give the regularity of the value functions \(u_{E}(t,x)\) and \(u_{A}(t,x)\) of the European option and the American option respectively. PubDate: 2017-10-01 DOI: 10.1007/s00245-016-9350-8 Issue No:Vol. 76, No. 2 (2017)

Authors:Paolo Acquistapace; Francesco Bartaloni Pages: 323 - 373 Abstract: Abstract We consider an optimal control problem arising in the context of economic theory of growth, on the lines of the works by Skiba and Askenazy–Le Van. The framework of the model is intertemporal infinite horizon utility maximization. The dynamics involves a state variable representing total endowment of the social planner or average capital of the representative dynasty. From the mathematical viewpoint, the main features of the model are the following: (i) the dynamics is an increasing, unbounded and not globally concave function of the state; (ii) the state variable is subject to a static constraint; (iii) the admissible controls are merely locally integrable in the right half-line. Such assumptions seem to be weaker than those appearing in most of the existing literature. We give a direct proof of the existence of an optimal control for any initial capital \(k_{0}\ge 0\) and we carry on a qualitative study of the value function; moreover, using dynamic programming methods, we show that the value function is a continuous viscosity solution of the associated Hamilton–Jacobi–Bellman equation. PubDate: 2017-10-01 DOI: 10.1007/s00245-016-9353-5 Issue No:Vol. 76, No. 2 (2017)

Authors:Lorena Bociu; Jean-Paul Zolésio Pages: 375 - 398 Abstract: Abstract We consider the wave equation with Dirichlet–Neumann boundary conditions on a family of perturbed domains \(\Omega _s\) . We discuss the shape differentiability analysis associated with the above mentioned problem, namely the existence of strong material and shape derivatives of the solution, and the rendering of the new wave problem whose solution is given by the shape derivative. The study shows that the Neumann boundary conditions completely change the focus and strategy involved in the shape differentiability analysis, in comparison to the case of the wave equation with purely Dirichlet boundary conditions. In this paper we show that for the existence of weak material derivative, the classical sensitivity analysis of the state can be bypassed by using parameter differentiability of a functional expressed in the form of Min–Max of a convex–concave Lagrangian with saddle point. Then we analyze the strong material derivative via a brute force estimate on the differential quotient, using known regularity results on the solution of the wave problem. PubDate: 2017-10-01 DOI: 10.1007/s00245-016-9354-4 Issue No:Vol. 76, No. 2 (2017)

Authors:Mo Chen Pages: 399 - 414 Abstract: Abstract In this paper, we obtain the existence of time optimal control of the Korteweg-de Vries-Burgers equation in a bounded domain with control acting locally in a subset. Moreover, we prove that any time optimal control satisfies the bang–bang property. PubDate: 2017-10-01 DOI: 10.1007/s00245-016-9355-3 Issue No:Vol. 76, No. 2 (2017)

Authors:Stefan Ankirchner; Christophette Blanchet-Scalliet; Monique Jeanblanc Pages: 415 - 428 Abstract: Abstract We consider the problem of maximizing the expected amount of time an exponential martingale spends above a constant threshold up to a finite time horizon. We assume that at any time the volatility of the martingale can be chosen to take any value between \(\sigma _1\) and \(\sigma _2\) , where \(0 < \sigma _1 < \sigma _2\) . The optimal control consists in choosing the minimal volatility \(\sigma _1\) when the process is above the threshold, and the maximal volatility if it is below. We give a rigorous proof using classical verification and provide integral formulas for the maximal expected occupation time above the threshold. PubDate: 2017-10-01 DOI: 10.1007/s00245-016-9356-2 Issue No:Vol. 76, No. 2 (2017)

Authors:Pei Pei; Mohammad A. Rammaha; Daniel Toundykov Pages: 429 - 464 Abstract: Abstract This study addresses well-posedness of a Mindlin–Timoshenko (MT) plate model that incorporates nonlinear viscous damping and nonlinear source term in Neumann boundary conditions. The main results verify local and global existence of solutions as well as their continuous dependence on the initial data in appropriate function spaces. Along with (Pei et al. in J Math Anal Appl 418(2):535–568, 2014, in Nonlinear Anal 105:62–85, 2014) this work completes the fundamental well-posedness theory for MT plates under the interplay of damping and source terms acting either in the interior or on the boundary of the plate. PubDate: 2017-10-01 DOI: 10.1007/s00245-016-9357-1 Issue No:Vol. 76, No. 2 (2017)

Authors:Jia Liu; Zhiping Chen; Abdel Lisser; Zhujia Xu Abstract: Abstract In this paper, we consider both one-period and multi-period distributionally robust mean-CVaR portfolio selection problems. We adopt an uncertainty set which considers the uncertainties in terms of both the distribution and the first two order moments. We use the parametric method and the dynamic programming technique to come up with the closed-form optimal solutions for both the one-period and the multi-period robust portfolio selection problems. Finally, we show that our approaches are efficient when compared with both normal based portfolio selection models, and robust approaches based on known moments. PubDate: 2017-10-13 DOI: 10.1007/s00245-017-9452-y

Authors:Pierluigi Colli; Gianni Gilardi; Gabriela Marinoschi; Elisabetta Rocca Abstract: Abstract In the present contribution we study the sliding mode control (SMC) problem for a diffuse interface tumor growth model coupling a viscous Cahn–Hilliard type equation for the phase variable with a reaction–diffusion equation for the nutrient. First, we prove the well-posedness and some regularity results for the state system modified by the state-feedback control law. Then, we show that the chosen SMC law forces the system to reach within finite time the sliding manifold (that we chose in order that the tumor phase remains constant in time). The feedback control law is added in the Cahn–Hilliard type equation and leads the phase onto a prescribed target \(\varphi ^*\) in finite time. PubDate: 2017-10-11 DOI: 10.1007/s00245-017-9451-z

Authors:Mircea Sofonea Abstract: Abstract The present paper represents a continuation of Migórski et al. (J Elast 127:151–178, 2017). There, the analysis of a new class of elliptic variational–hemivariational inequalities in reflexive Banach spaces, including existence and convergence results, was provided. An inequality in the class is governed by a nonlinear operator, a convex set of constraints and two nondifferentiable functionals, among which at least one is convex. In the current paper we complete this study with new results, including a convergence result with respect the set of constraints. Then we formulate two optimal control problems for which we prove the existence of optimal pairs, together with some convergence results. Finally, we exemplify our results in the study of a one-dimensional mathematical model which describes the equilibrium of an elastic rod in unilateral contact with a foundation, under the action of a body force. PubDate: 2017-09-22 DOI: 10.1007/s00245-017-9450-0

Authors:Jukka Lempa Abstract: Abstract We study an optimal investment problem with multiple entries and forced exits. A closed form solution of the optimisation problem is presented for general underlying diffusion dynamics and a general running payoff function in the case when forced exits occur on the jump times of a Poisson process. Furthermore, we allow the investment opportunity to be subject to the risk of a catastrophe that can occur at the jumps of the Poisson process. More precisely, we attach IID Bernoulli trials to the jump times and if the trial fails, no further re-entries are allowed. Interestingly, we find in the general case that the optimal investment threshold is independent of the success probability is the Bernoulli trials. The results are illustrated with explicit examples. PubDate: 2017-09-20 DOI: 10.1007/s00245-017-9449-6

Authors:Vincent Calvez; Thomas O. Gallouët Abstract: Abstract We investigate gradient flows of some homogeneous functionals in \(\mathbb R^N\) , arising in the Lagrangian approximation of systems of self-interacting and diffusing particles. We focus on the case of negative homogeneity. In the case of strong self-interaction (super critical case), the functional possesses a cone of negative energy. It is immediate to see that solutions with negative energy at some time become singular in finite time, meaning that a subset of particles concentrate at a single point. Here, we establish that all solutions become singular in finite time, in the super critical case, for the class of functionals under consideration. The paper is completed with numerical simulations illustrating the striking non linear dynamics when initial data have positive energy. PubDate: 2017-09-12 DOI: 10.1007/s00245-017-9443-z

Authors:K. Kadlec; B. Maslowski Abstract: In this paper, controlled linear stochastic evolution equations driven by square integrable Lévy processes are studied in the Hilbert space setting. The control operator may be unbounded which makes the results obtained in the abstract setting applicable to parabolic SPDEs with boundary or point control. The first part contains some preliminary technical results, notably a version of Itô formula which is applicable to weak/mild solutions of controlled equations. In the second part, the ergodic control problem is solved: The feedback form of the optimal control and the formula for the optimal cost are found. As examples, various parabolic type controlled SPDEs are studied. PubDate: 2017-09-11 DOI: 10.1007/s00245-017-9447-8

Authors:Carlo Orrieri; Gianmario Tessitore; Petr Veverka Abstract: Abstract We present a version of the stochastic maximum principle (SMP) for ergodic control problems. In particular we give necessary (and sufficient) conditions for optimality for controlled dissipative systems in finite dimensions. The strategy we employ is mainly built on duality techniques. We are able to construct a dual process for all positive times via the analysis of a suitable class of perturbed linearized forward equations. We show that such a process is the unique bounded solution to a backward SDE on infinite horizon from which we can write a version of the SMP. PubDate: 2017-09-06 DOI: 10.1007/s00245-017-9448-7

Authors:William M. McEneaney Abstract: Abstract A stochastic control representation for solution of the Schrödinger equation is obtained, utilizing complex-valued diffusion processes. The Maslov dequantization is employed, where the domain is complex-valued in the space variable. The notion of stationarity is utilized to relate the Hamilton–Jacobi form of the dequantized Schrödinger equation to its stochastic control representation. Convexity is not required, and consequently, there is no restriction on the duration of the problem. Additionally, existence is reduced to a real-valued domain case. PubDate: 2017-08-29 DOI: 10.1007/s00245-017-9442-0

Authors:Parsiad Azimzadeh Abstract: Abstract We study a zero-sum stochastic differential game (SDG) in which one controller plays an impulse control while their opponent plays a stochastic control. We consider an asymmetric setting in which the impulse player commits to, at the start of the game, performing less than q impulses (q can be chosen arbitrarily large). In order to obtain the uniform continuity of the value functions, previous works involving SDGs with impulses assume the cost of an impulse to be decreasing in time. Our work avoids such restrictions by requiring impulses to occur at rational times. We establish that the resulting game admits a value, and in turn, the existence and uniqueness of viscosity solutions to an associated Hamilton-Jacobi-Bellman-Isaacs quasi-variational inequality. PubDate: 2017-08-29 DOI: 10.1007/s00245-017-9445-x

Authors:S. Adly; F. Nacry Abstract: Abstract In this paper, we study the existence of solutions for a time and state-dependent discontinuous nonconvex second order sweeping process with a multivalued perturbation. The moving set is assumed to be prox-regular, relatively ball-compact with a bounded variation. The perturbation of the normal cone is a scalarly upper semicontinuous convex valued multimapping satisfying a linear growth condition possibly time-dependent. As an application of the theoretical results, we investigate the theory of evolution quasi-variational inequalities. PubDate: 2017-08-22 DOI: 10.1007/s00245-017-9446-9

Authors:Cung The Anh; Tran Minh Nguyet Abstract: Abstract We consider an optimal control problem for the 3D Navier–Stokes–Voigt equations in bounded domains, where the time needed to reach a desired state plays an essential role. We first prove the existence of optimal solutions, and then establish the first-order necessary optimality conditions and the second-order sufficient optimality conditions. PubDate: 2017-08-02 DOI: 10.1007/s00245-017-9441-1