Authors:Yousong Luo Pages: 151 - 173 Abstract: Abstract This paper deals with a class of optimal control problems governed by an initial-boundary value problem of a parabolic equation. The case of semi-linear boundary control is studied where the control is applied to the system via the Wentzell boundary condition. The differentiability of the state variable with respect to the control is established and hence a necessary condition is derived for the optimal solution in the case of both unconstrained and constrained problems. The condition is also sufficient for the unconstrained convex problems. A second order condition is also derived. PubDate: 2017-04-01 DOI: 10.1007/s00245-015-9326-0 Issue No:Vol. 75, No. 2 (2017)

Authors:G. M. Sklyar; P. Polak Pages: 175 - 192 Abstract: We consider an abstract Cauchy problem for a certain class of linear differential equations in Hilbert space. We obtain a criterion for stability on some dense subsets of the state space of the \(C_0\) -semigroups in terms of location of eigenvalues of their infinitesimal generators (so-called polynomial stability). We apply this result to analysis of stability and stabilizability of special class of neutral type systems with distributed delay. PubDate: 2017-04-01 DOI: 10.1007/s00245-015-9327-z Issue No:Vol. 75, No. 2 (2017)

Authors:Nikolaos S. Papageorgiou; Vicenţiu D. Rădulescu; Dušan D. Repovš Pages: 193 - 228 Abstract: Abstract We consider parametric equations driven by the sum of a p-Laplacian and a Laplace operator (the so-called (p, 2)-equations). We study the existence and multiplicity of solutions when the parameter \(\lambda >0\) is near the principal eigenvalue \(\hat{\lambda }_1(p)>0\) of \((-\Delta _p,W^{1,p}_{0}(\Omega ))\) . We prove multiplicity results with precise sign information when the near resonance occurs from above and from below of \(\hat{\lambda }_1(p)>0\) . PubDate: 2017-04-01 DOI: 10.1007/s00245-016-9330-z Issue No:Vol. 75, No. 2 (2017)

Authors:Feng-Qin Zhang; Rong Liu; Yuming Chen Pages: 229 - 251 Abstract: Abstract In this paper, we investigate a periodic food chain model with harvesting, where the predators have size structures and are described by first-order partial differential equations. First, we establish the existence of a unique non-negative solution by using the Banach fixed point theorem. Then, we provide optimality conditions by means of normal cone and adjoint system. Finally, we derive the existence of an optimal strategy by means of Ekeland’s variational principle. Here the objective functional represents the net economic benefit yielded from harvesting. PubDate: 2017-04-01 DOI: 10.1007/s00245-016-9331-y Issue No:Vol. 75, No. 2 (2017)

Authors:Fatima Zahra Mokkedem; Xianlong Fu Pages: 253 - 283 Abstract: Abstract In this paper, approximate controllability for a class of infinite-delayed semilinear stochastic systems in \(L_p\) space ( \(2<p<\infty \) ) is studied. The fundamental solution’s theory is used to describe the mild solution which is obtained by using the Banach fixed point theorem. In this way the approximate controllability result is then obtained by assuming that the corresponding deterministic linear system is approximately controllable via the so-called the resolvent condition. An application to a Volterra stochastic equation is also provided to illustrate the obtained results. PubDate: 2017-04-01 DOI: 10.1007/s00245-016-9332-x Issue No:Vol. 75, No. 2 (2017)

Authors:Joscha Diehl; Peter K. Friz; Paul Gassiat Pages: 285 - 315 Abstract: Abstract We study a class of controlled differential equations driven by rough paths (or rough path realizations of Brownian motion) in the sense of Lyons. It is shown that the value function satisfies a HJB type equation; we also establish a form of the Pontryagin maximum principle. Deterministic problems of this type arise in the duality theory for controlled diffusion processes and typically involve anticipating stochastic analysis. We make the link to old work of Davis and Burstein (Stoch Stoch Rep 40:203–256, 1992) and then prove a continuous-time generalization of Roger’s duality formula [SIAM J Control Optim 46:1116–1132, 2007]. The generic case of controlled volatility is seen to give trivial duality bounds, and explains the focus in Burstein–Davis’ (and this) work on controlled drift. Our study of controlled rough differential equations also relates to work of Mazliak and Nourdin (Stoch Dyn 08:23, 2008). PubDate: 2017-04-01 DOI: 10.1007/s00245-016-9333-9 Issue No:Vol. 75, No. 2 (2017)

Authors:Xianping Guo; Yonghui Huang; Yi Zhang Pages: 317 - 341 Abstract: Abstract This paper studies the constrained (nonhomogeneous) continuous-time Markov decision processes on the finite horizon. The performance criterion to be optimized is the expected total reward on the finite horizon, while N constraints are imposed on similar expected costs. Introducing the appropriate notion of the occupation measures for the concerned optimal control problem, we establish the following under some suitable conditions: (a) the class of Markov policies is sufficient; (b) every extreme point of the space of performance vectors is generated by a deterministic Markov policy; and (c) there exists an optimal Markov policy, which is a mixture of no more than \(N+1\) deterministic Markov policies. PubDate: 2017-04-01 DOI: 10.1007/s00245-016-9352-6 Issue No:Vol. 75, No. 2 (2017)

Authors:Beniamin Bogosel Pages: 1 - 25 Abstract: Abstract We study the continuity of the Steklov spectrum on variable domains with respect to the Hausdorff convergence. A key point of the article is understanding the behaviour of the traces of Sobolev functions on moving boundaries of sets satisfying an uniform geometric condition. As a consequence, we are able to prove existence results for shape optimization problems regarding the Steklov spectrum in the family of sets satisfying a \(\varepsilon \) -cone condition and in the family of convex sets. PubDate: 2017-02-01 DOI: 10.1007/s00245-015-9321-5 Issue No:Vol. 75, No. 1 (2017)

Authors:Jerry L. Bona; Hongqiu Chen; Ohannes Karakashian Pages: 27 - 53 Abstract: Abstract The present study is concerned with systems $$\begin{aligned} \left\{ \begin{array}{ll} &{} \frac{\partial u}{\partial t} +\frac{\partial ^3 u}{\partial x^3} + \frac{\partial }{\partial x}P(u, v)=0,\\ &{} \frac{\partial v}{\partial t} +\frac{\partial ^3 v}{\partial x^3} + \frac{\partial }{\partial x}Q(u, v)=0, \end{array}\right. \end{aligned}$$ of Korteweg–de Vries type, coupled through their nonlinear terms. Here, \(u = u(x,t)\) and \(v = v(x,t)\) are real-valued functions of a real spatial variable x and a real temporal variable t. The nonlinearities P and Q are homogeneous, quadratic polynomials with real coefficients \(A,B,\ldots \) , viz. $$\begin{aligned} P(u,v)=Au^2+Buv+Cv^2, \qquad Q(u,v)=Du^2+Euv+Fv^2, \end{aligned}$$ in the dependent variables u and v. A satisfactory theory of local well-posedness is in place for such systems. Here, attention is drawn to their solitary-wave solutions. Special traveling waves termed proportional solitary waves are introduced and determined. Under the same conditions developed earlier for global well-posedness, stability criteria are obtained for these special, traveling-wave solutions. PubDate: 2017-02-01 DOI: 10.1007/s00245-015-9322-4 Issue No:Vol. 75, No. 1 (2017)

Authors:I. Abraham; R. Abraham; M. Bergounioux; G. Carlier Pages: 55 - 73 Abstract: Abstract In this article, we focus on tomographic reconstruction. The problem is to determine the shape of the interior interface using a tomographic approach while very few X-ray radiographs are performed. We use a multi-marginal optimal transport approach. Preliminary numerical results are presented. PubDate: 2017-02-01 DOI: 10.1007/s00245-015-9323-3 Issue No:Vol. 75, No. 1 (2017)

Authors:Bernt Øksendal; Agnès Sulem Pages: 117 - 147 Abstract: Abstract A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities: The optimal terminal wealth \(X^*(T) : = X_{\varphi ^*}(T)\) of the problem to maximize the expected U-utility of the terminal wealth \(X_{\varphi }(T)\) generated by admissible portfolios \(\varphi (t); 0 \le t \le T\) in a market with the risky asset price process modeled as a semimartingale; The optimal scenario \(\frac{dQ^*}{dP}\) of the dual problem to minimize the expected V-value of \(\frac{dQ}{dP}\) over a family of equivalent local martingale measures Q, where V is the convex conjugate function of the concave function U. In this paper we consider markets modeled by Itô-Lévy processes. In the first part we use the maximum principle in stochastic control theory to extend the above relation to a dynamic relation, valid for all \(t \in [0,T]\) . We prove in particular that the optimal adjoint process for the primal problem coincides with the optimal density process, and that the optimal adjoint process for the dual problem coincides with the optimal wealth process; \(0 \le t \le T\) . In the terminal time case \(t=T\) we recover the classical duality connection above. We get moreover an explicit relation between the optimal portfolio \(\varphi ^*\) and the optimal measure \(Q^*\) . We also obtain that the existence of an optimal scenario is equivalent to the replicability of a related T-claim. In the second part we present robust (model uncertainty) versions of the optimization problems in (i) and (ii), and we prove a similar dynamic relation between them. In particular, we show how to get from the solution of one of the problems to the other. We illustrate the results with explicit examples. PubDate: 2017-02-01 DOI: 10.1007/s00245-016-9329-5 Issue No:Vol. 75, No. 1 (2017)

Abstract: Abstract In this work, we introduce and study a Prigozhin model for growing sandpile with mixed boundary conditions. For theoretical analysis we use semi-group theory and the numerical part is based on a duality approach. PubDate: 2017-03-21

Authors:Harbir Antil; Shawn W. Walker Abstract: Abstract Controlling the shapes of surfaces provides a novel way to direct self-assembly of colloidal particles on those surfaces and may be useful for material design. This motivates the investigation of an optimal control problem for surface shape in this paper. Specifically, we consider an objective (tracking) functional for surface shape with the prescribed mean curvature equation in graph form as a state constraint. The control variable is the prescribed curvature. We prove existence of an optimal control, and using improved regularity estimates, we show sufficient differentiability to make sense of the first order optimality conditions. This allows us to rigorously compute the gradient of the objective functional for both the continuous and discrete (finite element) formulations of the problem. Numerical results are shown to illustrate the minimizers and optimal controls on different domains. PubDate: 2017-03-16 DOI: 10.1007/s00245-017-9407-3

Authors:F. D. Araruna; B. M. R. Calsavara; E. Fernández-Cara Abstract: Abstract We analyze a control problem for a phase field system modeling the solidification process of materials that allow two different types of crystallization coupled to a Navier–Stokes system and a nonlinear heat equation, with a reduced number of controls. We prove that this system is locally exactly controllable to suitable trajectories, with controls acting only on the motion and heat equations. PubDate: 2017-03-04 DOI: 10.1007/s00245-017-9406-4

Authors:C. A. Bortot; M. M. Cavalcanti; V. N. Domingos Cavalcanti; P. Piccione Abstract: Abstract The Klein Gordon equation subject to a nonlinear and locally distributed damping, posed in a complete and non compact n dimensional Riemannian manifold \((\mathcal {M}^n,\mathbf {g})\) without boundary is considered. Let us assume that the dissipative effects are effective in \((\mathcal {M}\backslash \Omega ) \cup (\Omega \backslash V)\) , where \(\Omega \) is an arbitrary open bounded set with smooth boundary. In the present article we introduce a new class of non compact Riemannian manifolds, namely, manifolds which admit a smooth function f, such that the Hessian of f satisfies the pinching conditions (locally in \(\Omega \) ), for those ones, there exist a finite number of disjoint open subsets \( V_k\) free of dissipative effects such that \(\bigcup _k V_k \subset V\) and for all \(\varepsilon >0\) , \(meas(V)\ge meas(\Omega )-\varepsilon \) , or, in other words, the dissipative effect inside \(\Omega \) possesses measure arbitrarily small. It is important to be mentioned that if the function f satisfies the pinching conditions everywhere, then it is not necessary to consider dissipative effects inside \(\Omega \) . PubDate: 2017-03-03 DOI: 10.1007/s00245-017-9405-5

Authors:Weiwei Hu Abstract: 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: 2017-03-01 DOI: 10.1007/s00245-017-9404-6

Authors:Jingrui Sun; Jiongmin Yong Abstract: 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: 2017-02-23 DOI: 10.1007/s00245-017-9402-8

Authors:Giuseppe Buttazzo; Thierry Champion; Luigi De Pascale Abstract: 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: 2017-02-18 DOI: 10.1007/s00245-017-9403-7

Authors:J. Doll; P. Dupuis; P. Nyquist Abstract: 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: 2017-02-08 DOI: 10.1007/s00245-017-9401-9