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:O. L. V. Costa; F. Dufour Abstract: Abstract This papers deals with the zero-sum game with a discounted reward criterion for piecewise deterministic Markov process (PDMPs) in general Borel spaces. The two players can act on the jump rate and transition measure of the process, with the decisions being taken just after a jump of the process. The goal of this paper is to derive conditions for the existence of min–max strategies for the infinite horizon total expected discounted reward function, which is composed of running and boundary parts. The basic idea is, by using the special features of the PDMPs, to re-write the problem via an embedded discrete-time Markov chain associated to the PDMP and re-formulate the problem as a discrete-stage zero sum game problem. PubDate: 2017-04-25 DOI: 10.1007/s00245-017-9416-2

Authors:Michele Colturato Abstract: Abstract We prove existence and regularity for the solutions to a Cahn–Hilliard system describing the phenomenon of phase separation for a material contained in a bounded and regular domain. Since the first equation of the system is perturbed by the presence of an additional maximal monotone operator, we show our results using suitable regularization of the nonlinearities of the problem and performing some a priori estimates which allow us to pass to the limit thanks to compactness and monotonicity arguments. Next, under further assumptions, we deduce a continuous dependence estimate whence the uniqueness property is also achieved. Then, we consider the related sliding mode control (SMC) problem and show that the chosen SMC law forces a suitable linear combination of the temperature and the phase to reach a given (space-dependent) value within finite time. PubDate: 2017-04-18 DOI: 10.1007/s00245-017-9415-3

Authors:Harald Garcke; Kei Fong Lam; Elisabetta Rocca Abstract: Abstract We consider an optimal control problem for a diffuse interface model of tumor growth. The state equations couples a Cahn–Hilliard equation and a reaction-diffusion equation, which models the growth of a tumor in the presence of a nutrient and surrounded by host tissue. The introduction of cytotoxic drugs into the system serves to eliminate the tumor cells and in this setting the concentration of the cytotoxic drugs will act as the control variable. Furthermore, we allow the objective functional to depend on a free time variable, which represents the unknown treatment time to be optimized. As a result, we obtain first order necessary optimality conditions for both the cytotoxic concentration and the treatment time. PubDate: 2017-04-13 DOI: 10.1007/s00245-017-9414-4

Authors:Géraldine Bouveret; Jean-François Chassagneux Abstract: Abstract In a Markovian framework, we consider the problem of finding the minimal initial value of a controlled process allowing to reach a stochastic target with a given level of expected loss. This question arises typically in approximate hedging problems. The solution to this problem has been characterised by Bouchard et al. (SIAM J Control Optim 48(5):3123–3150, 2009) and is known to solve an Hamilton–Jacobi–Bellman PDE with discontinuous operator. In this paper, we prove a comparison theorem for the corresponding PDE by showing first that it can be rewritten using a continuous operator, in some cases. As an application, we then study the quantile hedging price of Bermudan options in the non-linear case, pursuing the study initiated in Bouchard et al. (J Financial Math 7(1):215–235, 2016). PubDate: 2017-04-10 DOI: 10.1007/s00245-017-9413-5

Authors:R. A. Cipolatti; I.-S. Liu; L. A. Palermo; M. A. Rincon; R. M. S. Rosa Abstract: Abstract A Stokes-type problem for a viscoelastic model of salt rocks is considered, and existence, uniqueness and regularity are investigated in the scale of \(L^2\) -based Sobolev spaces. The system is transformed into a generalized Stokes problem, and the proper conditions on the parameters of the model that guarantee that the system is uniformly elliptic are given. Under those conditions, existence, uniqueness and low-order regularity are obtained under classical regularity conditions on the data, while higher-order regularity is proved under less stringent conditions than classical ones. Explicit estimates for the solution in terms of the data are given accordingly. PubDate: 2017-04-04 DOI: 10.1007/s00245-017-9411-7

Authors:Erhan Bayraktar; Zhou Zhou Abstract: Abstract On a filtered probability space \((\Omega ,\mathcal {F},P,\mathbb {F}=(\mathcal {F}_t)_{t=0,\ldots ,T})\) , we consider stopping games \(\overline{V}:=\inf _{{\varvec{\rho }}\in \mathbb {T}^{ii}}\sup _{\tau \in \mathcal {T}}\mathbb {E}[U({\varvec{\rho }}(\tau ),\tau )]\) and \(\underline{V}:=\sup _{{\varvec{\tau }}\in \mathbb {T}^i}\inf _{\rho \in \mathcal {T}}\mathbb {E}[U(\rho ,{\varvec{\tau }}(\rho ))]\) in discrete time, where U(s, t) is \(\mathcal {F}_{s\vee t}\) -measurable instead of \(\mathcal {F}_{s\wedge t}\) -measurable as is assumed in the literature on Dynkin games, \(\mathcal {T}\) is the set of stopping times, and \(\mathbb {T}^i\) and \(\mathbb {T}^{ii}\) are sets of mappings from \(\mathcal {T}\) to \(\mathcal {T}\) satisfying certain non-anticipativity conditions. We will see in an example that there is no room for stopping strategies in classical Dynkin games unlike the new stopping game we are introducing. We convert the problems into an alternative Dynkin game, and show that \(\overline{V}=\underline{V}=V\) , where V is the value of the Dynkin game. We also get optimal \({\varvec{\rho }}\in \mathbb {T}^{ii}\) and \({\varvec{\tau }}\in \mathbb {T}^i\) for \(\overline{V}\) and \(\underline{V}\) respectively. PubDate: 2017-04-04 DOI: 10.1007/s00245-017-9412-6

Authors:Andaluzia Matei; Sorin Micu Abstract: Abstract We consider the contact between an elastic body and a deformable foundation. Firstly, we introduce a mathematical model for this phenomenon by means of a normal compliance contact condition associated with a friction law. Then, we propose a variational formulation of the model in a form of a quasi-variational inequality governed by a non-differentiable functional and we briefly discuss its well-possedness. Nextly, we address an optimal control problem related to this model in order to led the displacement field as close as possible to a given target by acting with a localized boundary control. By using some mollifiers of the normal compliance functions, we introduce a regularized model which allows us to establish an optimality condition. Finally, by means of asymptotic analysis tools, we show that the solutions of the regularized optimal control problems converge to a solution of the initial optimal control problem. PubDate: 2017-03-30 DOI: 10.1007/s00245-017-9410-8

Authors:Viorel Barbu; Zdzisław Brzeźniak; Luciano Tubaro Abstract: Abstract Existence and uniqueness of solutions to stochastic differential equation \(dX-\text {div}\,a(\nabla X)\,dt=\sum _{j=1}^N(b_j\cdot \nabla X)\circ d\beta _j\) in \((0,T)\times \mathcal O\) ; \(X(0,\xi )=x(\xi )\) , \(\xi \in \mathcal O\) , \(X=0\) on \((0,T)\times \partial \mathcal O\) is studied. Here \(\mathcal O\) is a bounded and open domain of \(\mathbb R^d\) , \(d\ge 1\) , \(\{b_j\}\) is a divergence free vector field, \(a:[0,T]\times \mathcal O\times \mathbb R^d\rightarrow \mathbb R^d\) is a continuous and monotone mapping of subgradient type and \(\{\beta _j\}\) are independent Brownian motions in a probability space \((\Omega ,\mathcal F,\mathbb P)\) . The weak solution is defined via stochastic optimal control problem. PubDate: 2017-03-27 DOI: 10.1007/s00245-017-9409-1

Authors:N. Igbida; F. Karami; S. Ouaro; U. Traoré 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 DOI: 10.1007/s00245-017-9408-2

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