Authors:Radouen Ghanem; Ibtissam Nouri Pages: 465 - 500 Abstract: We consider an optimal control problem for the obstacle problem with an elliptic polyharmonic obstacle problem of order 2m, where the obstacle function is assumed to be the control. We use a Moreau–Yosida approximate technique to introduce a family of problems governed by variational equations. Then, we prove optimal solutions existence and give an approximate optimality system and convergence results by passing to the limit in this system. PubDate: 2017-12-01 DOI: 10.1007/s00245-016-9358-0 Issue No:Vol. 76, No. 3 (2017)

Authors:Xianggang Lu; George Yin; Qing Zhang; Caojin Zhang; Xianping Guo Pages: 501 - 533 Abstract: This paper is concerned with modeling, analysis, and numerical methods for stochastic optimal control of an illiquid stock position build-up. The stock price model is based on a geometric Brownian motion formulation, in which the drift is allowed to be purchase-rate dependent to capture the “price impact” of heavy share accumulation over time. The expected fund (or capital) availability has an upper bound. A Lagrange multiplier method is used to treat the constrained control problem. The stochastic control problem is analyzed and a verification theorem is developed. Although optimality is proved, a closed-form solution is virtually impossible to obtain. As a viable alternative, approximation schemes are developed, which consist of inner and outer approximations. The inner approximation is a numerical procedure for obtaining optimal strategies based on a fixed parameter of the Lagrange multiplier. The outer approximation is a stochastic approximation algorithm for obtaining the optimal Lagrange multiplier. Convergence analysis is provided for both the inner and outer approximations. Finally, numerical examples are provided to illustrate our results. PubDate: 2017-12-01 DOI: 10.1007/s00245-016-9359-z Issue No:Vol. 76, No. 3 (2017)

Authors:Igor Kukavica; Amjad Tuffaha; Vlad Vicol Pages: 535 - 563 Abstract: We address the local existence and uniqueness of solutions for the 3D Euler equations with a free interface. We prove the local well-posedness in the rotational case when the initial datum \(u_0\) satisfies \(u_0\in H^{2.5+\delta }\) and , where \(\delta >0\) is arbitrarily small, under the Taylor condition on the pressure. PubDate: 2017-12-01 DOI: 10.1007/s00245-016-9360-6 Issue No:Vol. 76, No. 3 (2017)

Authors:Guillaume Carlier; Xavier Dupuis Pages: 565 - 592 Abstract: The principal-agent problem in economics leads to variational problems subject to global constraints of b-convexity on the admissible functions, capturing the so-called incentive-compatibility constraints. Typical examples are minimization problems subject to a convexity constraint. In a recent pathbreaking article, Figalli et al. (J Econ Theory 146(2):454–478, 2011) identified conditions which ensure convexity of the principal-agent problem and thus raised hope on the development of numerical methods. We consider special instances of projections problems over b-convex functions and show how they can be solved numerically using Dykstra’s iterated projection algorithm to handle the b-convexity constraint in the framework of (Figalli et al. in J Econ Theory 146(2):454–478, 2011). Our method also turns out to be simple for convex envelope computations. PubDate: 2017-12-01 DOI: 10.1007/s00245-016-9361-5 Issue No:Vol. 76, No. 3 (2017)

Authors:P. Mondal; S. K. Neogy; S. Sinha; D. Ghorui Pages: 593 - 619 Abstract: In this article, we revisit the applications of principal pivot transform and its generalization for a rectangular matrix (in the context of vertical linear complementarity problem) to solve some structured classes of zero-sum two-person discounted semi-Markov games with finitely many states and actions. The single controller semi-Markov games have been formulated as a linear complementarity problem and solved using a stepwise principal pivoting algorithm. We provide a sufficient condition for such games to be completely mixed. The concept of switching controller semi-Markov games is introduced and we prove the ordered field property and the existence of stationary optimal strategies for such games. Moreover, such games are formulated as a vertical linear complementarity problem and have been solved using a stepwise generalized principal pivoting algorithm. Sufficient conditions are also given for such games to be completely mixed. For both these classes of games, some properties analogous to completely mixed matrix games, are established. PubDate: 2017-12-01 DOI: 10.1007/s00245-016-9362-4 Issue No:Vol. 76, No. 3 (2017)

Authors:Nikolaos S. Papageorgiou; Vicenţiu D. Rădulescu Pages: 621 - 639 Abstract: We consider a nonlinear Dirichlet problem driven by the sum of p-Laplacian and a Laplacian (a (p, 2)-equation) which is resonant at \(\pm \infty \) with respect to the principal eigenvalue \(\hat{\lambda }_1(p)\) of \((-\Delta _p,W^{1,p}_{0}(\Omega ))\) and resonant at zero with respect to any nonprincipal eigenvalue of \((-\Delta ,H^1_0(\Omega ))\) . At \(\pm \infty \) the resonance occurs from the right of \(\hat{\lambda }_1(p)\) and so the energy functional of the problem is indefinite. Using critical groups, we show that the problem has at least one nontrivial smooth solution. The result complements the recent work of Papageorgiou and Rădulescu (Appl Math Optim 69:393–430, 2014), where resonant (p, 2)-equations were examined with the resonance occurring from the left of \(\hat{\lambda }_1(p)\) (coercive problem). PubDate: 2017-12-01 DOI: 10.1007/s00245-016-9363-3 Issue No:Vol. 76, No. 3 (2017)

Authors:Filippo Dell’Oro; Vittorino Pata Pages: 641 - 655 Abstract: We discuss the parallel between the third-order Moore–Gibson–Thompson equation $$\begin{aligned} {\partial _{ttt}} u + \alpha {\partial _{tt}}u-\beta \Delta {\partial _t} u - \gamma \Delta u =0 \end{aligned}$$ depending on the parameters \(\alpha ,\beta ,\gamma >0,\) and the equation of linear viscoelasticity $$\begin{aligned} \partial _{tt}u(t) - \kappa (0)\Delta u(t) - \int _{0}^\infty \kappa ^{\prime }(s)\Delta u(t-s)\,\mathrm{d}s=0 \end{aligned}$$ for the particular choice of the exponential kernel $$\begin{aligned} \kappa (s) = a \mathrm{e}^{-b s} + c \end{aligned}$$ with \(a,b,c>0\) . In particular, the latter model is shown to exhibit a preservation of regularity for a certain class of initial data, which is unexpected in presence of a general memory kernel \(\kappa \) . PubDate: 2017-12-01 DOI: 10.1007/s00245-016-9365-1 Issue No:Vol. 76, No. 3 (2017)

Authors:Vanja Nikolić; Barbara Kaltenbacher Pages: 261 - 301 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: 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: 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:Mo Chen Pages: 399 - 414 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:Pei Pei; Mohammad A. Rammaha; Daniel Toundykov Pages: 429 - 464 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:Guy Bouchitté; Ilaria Fragalà; Ilaria Lucardesi Abstract: We show that the compliance functional in elasticity is differentiable with respect to horizontal variations of the load term, when the latter is given by a possibly concentrated measure; moreover, we provide an integral representation formula for the derivative as a linear functional of the deformation vector field. The result holds true as well for the p-compliance in the scalar case of conductivity. Then we study the limit problem as \(p \rightarrow + \infty \) , which corresponds to differentiate the Wasserstein distance in optimal mass transportation with respect to horizontal perturbations of the two marginals. Also in this case, we obtain an existence result for the derivative, and we show that it is found by solving a minimization problem over the family of all optimal transport plans. When the latter contains only one element, we prove that the derivative of the p-compliance converges to the derivative of the Wasserstein distance in the limit as \(p \rightarrow + \infty \) . PubDate: 2017-11-11 DOI: 10.1007/s00245-017-9455-8

Authors:Zhang Binlin; Alessio Fiscella; Sihua Liang Abstract: In this paper we study a class of critical Kirchhoff type equations involving the fractional p–Laplacian operator, that is $$\begin{aligned} \begin{array}{ll} \displaystyle M\left( \iint _{{\mathbb {R}}^{2N}}\frac{ u(x)-u(y) ^p}{ x-y ^{N+ps}}dxdy\right) (-\Delta )_p^{s} u {=} \lambda w(x) u ^{q-2}u + u ^{ p_s^{*}-2 }u,\quad x\in {\mathbb {R}}^N, \end{array} \end{aligned}$$ where \((-\,\Delta )^s_p\) is the fractional p–Laplacian operator with \(0<s<1<p<\infty \) , dimension \(N>ps\) , \(1<q<p^{*}_{s}\) , \(p^{*}_s\) is the critical exponent of the fractional Sobolev space \(W^{s,p}({\mathbb {R}}^N)\) , \(\lambda \) is a positive parameter, M is a non-negative function while w is a positive weight. By exploiting Kajikiya’s new version of the symmetric mountain pass lemma, we establish the existence of infinitely many solutions which tend to zero under a suitable value of \(\lambda \) . The main feature and difficulty of our equations is the fact that the Kirchhoff term M is zero at zero, that is the equation is degenerate. To our best knowledge, our results are new even in the Laplacian and p–Laplacian cases. PubDate: 2017-11-08 DOI: 10.1007/s00245-017-9458-5

Authors:Denis Belomestny; Roland Hildebrand; John Schoenmakers Abstract: The optimal stopping problem arising in the pricing of American options can be tackled by the so called dual martingale approach. In this approach, a dual problem is formulated over the space of adapted martingales. A feasible solution of the dual problem yields an upper bound for the solution of the original primal problem. In practice, the optimization is performed over a finite-dimensional subspace of martingales. A sample of paths of the underlying stochastic process is produced by a Monte-Carlo simulation, and the expectation is replaced by the empirical mean. As a rule the resulting optimization problem, which can be written as a linear program, yields a martingale such that the variance of the obtained estimator can be large. In order to decrease this variance, a penalizing term can be added to the objective function of the pathwise optimization problem. In this paper, we provide a rigorous analysis of the optimization problems obtained by adding different penalty functions. In particular, a convergence analysis implies that it is better to minimize the empirical maximum instead of the empirical mean. Numerical simulations confirm the variance reduction effect of the new approach. PubDate: 2017-11-08 DOI: 10.1007/s00245-017-9454-9

Authors:Wenjun Liu; Weifan Zhao Abstract: In this paper, we study the well-posedness and asymptotic stability of a thermoelastic laminated beam with past history. For the system with structural damping, without any restriction on the speeds of wave propagations, we prove the exponential and polynomial stabilities which depend on the behavior of the kernel function of the history term, by using the perturbed energy method. For the system without structural damping, we prove the exponential and polynomial stabilities in case of equal speeds and lack of exponential stability in case of non-equal speeds by using the perturbed energy method and Gearhart–Herbst–Prüss–Huang theorem, respectively. Furthermore, the well-posedness of the system is also obtained by using Lumer–Philips theorem. PubDate: 2017-11-08 DOI: 10.1007/s00245-017-9460-y

Authors:D. Gómez; E. Pérez; A. V. Podolskii; T. A. Shaposhnikova Abstract: We address homogenization problems of variational inequalities for the p-Laplace operator in a domain of \(\mathbb {R}^n\) ( \(n\ge \) 3, \(p\in [2,n)\) ) periodically perforated by balls of radius \(O(\varepsilon ^\alpha )\) where \(\alpha >1\) and \(\varepsilon \) is the size of the period. The perforations are distributed along a \((n-1)\) -dimensional manifold \(\gamma \) , and we impose constraints for solutions and their fluxes (associated with the p-Laplacian) on the boundary of the perforations. These constraints imply that the solution is positive and that the flux is bounded from above by a negative, nonlinear monotonic function of the solution multiplied by a parameter \(\varepsilon ^{-\kappa }\) , \(\kappa \in \mathbb {R}\) and \(\varepsilon \) is a small parameter that we shall make to go to zero. We analyze different relations between the parameters \(p, \, n, \, \varepsilon , \, \alpha \) and \(\kappa \) , and obtain homogenized problems which are completely new in the literature even for the case \(p=2\) . PubDate: 2017-11-02 DOI: 10.1007/s00245-017-9453-x

Authors:Keying Ma; Tongjun Sun Abstract: In this paper, we consider a non-overlapping domain decomposition method for solving optimal boundary control problems governed by parabolic equations. The whole domain is divided into non-overlapping subdomains, and the global optimal boundary control problem is decomposed into the local problems in these subdomains. The integral mean method is utilized to present an explicit flux calculation on the inter-domain boundary in order to communicate the local problems on the interface between subdomains. We establish the fully parallel and discrete schemes for solving these local problems. A priori error estimates in \(L^2\) -norm are derived for the state, co-state and control variables. Finally, we present numerical experiments to show the validity of the schemes and verify the derived theoretical results. PubDate: 2017-10-28 DOI: 10.1007/s00245-017-9456-7

Authors:Jia Liu; Zhiping Chen; Abdel Lisser; Zhujia Xu 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: 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