Authors:C. A. Bortot; M. M. Cavalcanti; V. N. Domingos Cavalcanti; P. Piccione Pages: 219 - 265 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: 2018-10-01 DOI: 10.1007/s00245-017-9405-5 Issue No:Vol. 78, No. 2 (2018)

Authors:F. D. Araruna; B. M. R. Calsavara; E. Fernández-Cara Pages: 267 - 296 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: 2018-10-01 DOI: 10.1007/s00245-017-9406-4 Issue No:Vol. 78, No. 2 (2018)

Authors:Harbir Antil; Shawn W. Walker Pages: 297 - 328 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: 2018-10-01 DOI: 10.1007/s00245-017-9407-3 Issue No:Vol. 78, No. 2 (2018)

Authors:N. Igbida; F. Karami; S. Ouaro; U. Traoré Pages: 329 - 359 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: 2018-10-01 DOI: 10.1007/s00245-017-9408-2 Issue No:Vol. 78, No. 2 (2018)

Authors:Viorel Barbu; Zdzisław Brzeźniak; Luciano Tubaro Pages: 361 - 377 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: 2018-10-01 DOI: 10.1007/s00245-017-9409-1 Issue No:Vol. 78, No. 2 (2018)

Authors:Andaluzia Matei; Sorin Micu Pages: 379 - 401 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: 2018-10-01 DOI: 10.1007/s00245-017-9410-8 Issue No:Vol. 78, No. 2 (2018)

Authors:R. A. Cipolatti; I.-S. Liu; L. A. Palermo; M. A. Rincon; R. M. S. Rosa Pages: 403 - 456 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: 2018-10-01 DOI: 10.1007/s00245-017-9411-7 Issue No:Vol. 78, No. 2 (2018)

Authors:José A. Iglesias; Gwenael Mercier; Otmar Scherzer Abstract: We consider the flow of multiple particles in a Bingham fluid in an anti-plane shear flow configuration. The limiting situation in which the internal and applied forces balance and the fluid and particles stop flowing, that is, when the flow settles, is formulated as finding the optimal ratio between the total variation functional and a linear functional. The minimal value for this quotient is referred to as the critical yield number or, in analogy to Rayleigh quotients, generalized eigenvalue. This minimum value can in general only be attained by discontinuous, hence not physical, velocities. However, we prove that these generalized eigenfunctions, whose jumps we refer to as limiting yield surfaces, appear as rescaled limits of the physical velocities. Then, we show the existence of geometrically simple minimizers. Furthermore, a numerical method for the minimization is then considered. It is based on a nonlinear finite difference discretization, whose consistency is proven, and a standard primal-dual descent scheme. Finally, numerical examples show a variety of geometric solutions exhibiting the properties discussed in the theoretical sections. PubDate: 2018-10-10 DOI: 10.1007/s00245-018-9531-8

Authors:V. M. Veliov; P. T. Vuong Abstract: The paper presents new results about convergence of the gradient projection and the conditional gradient methods for abstract minimization problems on strongly convex sets. In particular, linear convergence is proved, although the objective functional does not need to be convex. Such problems arise, in particular, when a recently developed discretization technique is applied to optimal control problems which are affine with respect to the control. This discretization technique has the advantage to provide higher accuracy of discretization (compared with the known discretization schemes) and involves strongly convex constraints and possibly non-convex objective functional. The applicability of the abstract results is proved in the case of linear-quadratic affine optimal control problems. A numerical example is given, confirming the theoretical findings. PubDate: 2018-10-06 DOI: 10.1007/s00245-018-9528-3

Authors:Jingrui Sun Abstract: This paper is concerned with a linear quadratic (LQ, for short) optimal control problem with fixed terminal states and integral quadratic constraints. A Riccati equation with infinite terminal value is introduced, which is uniquely solvable and whose solution can be approximated by the solution for a suitable unconstrained LQ problem with penalized terminal state. Using results from duality theory, the optimal control is explicitly derived by solving the Riccati equation together with an optimal parameter selection problem. It turns out that the optimal control is a target-dependent feedback of the current state. Some examples are presented to illustrate the theory developed. PubDate: 2018-10-04 DOI: 10.1007/s00245-018-9532-7

Authors:Carole Louis-Rose; Mahamadi Warma Abstract: We investigate the controllability from the exterior of space-time fractional wave equations involving the Caputo time fractional derivative with the fractional Laplace operator subject to nonhomogeneous Dirichlet or Robin type exterior conditions. We prove that if \(1<\alpha < 2\) , \(0<s<1\) and \(\Omega \subset {{\mathbb {R}}}^N\) is a bounded Lipschitz domain, then the system $$\begin{aligned} {\left\{ \begin{array}{ll} {\mathbb {D}}_t^\alpha u+(-\Delta )^su=0\;\;&{} \text{ in } \;\;\Omega \times (0,T), \\ Bu=g\;&{} \text{ in } \; ({\mathbb {R}}^N\setminus \Omega )\times (0,T), \\ u(\cdot ,0)=u_0, \;\;\partial _tu(\cdot ,0)=u_1\;&{} \text{ in } \;\Omega , \end{array}\right. } \end{aligned}$$ is approximately controllable for any \(T>0\) , \((u_0,u_1)\in L^2(\Omega )\times {\mathbb {V}}_B^{-\frac{1}{\alpha }}\) and every \(g\in {\mathcal {D}}({\mathcal {O}}\times (0,T))\) , where \({\mathcal {O}}\subset ({\mathbb {R}}^N\setminus \Omega )\) is any non-empty open set in the case of the Dirichlet exterior condition \(Bu=u\) , and \({\mathcal {O}}\subseteq {\mathbb {R}}^N\setminus \Omega \) is any open set dense in \({\mathbb {R}}^N\setminus \Omega \) for the Robin exterior conditions \(Bu:=\mathcal N_su+\kappa u\) . Here, \({\mathcal {N}}_su\) is the nonlocal normal derivative of u and \({\mathbb {V}}_B^{-\frac{1}{\alpha }}\) denotes the dual of the domain of the fractional power of order \(\frac{1}{\alpha }\) of the realization in \(L^2(\Omega )\) of the operator \((-\Delta )^s\) with the zero (Dirichlet or Robin) exterior conditions \(Bu=0\) in \({{\mathbb {R}}}^N\setminus \Omega \) . PubDate: 2018-09-26 DOI: 10.1007/s00245-018-9530-9

Authors:Patrik Knopf; Jörg Weber Abstract: We consider the Vlasov–Poisson system that is equipped with an external magnetic field to describe the time evolution of the distribution function of a plasma. An optimal control problem where the external magnetic field is the control itself has already been investigated by Knopf (Calc Var 57:134, 2018). However, in real technical applications it will not be possible to choose the control field in such a general fashion as it will be induced by fixed field coils. In this paper we will use the fundamentals that were established by Knopf (Calc Var 57:134, 2018) to analyze an optimal control problem where the magnetic field is a superposition of the fields that are generated by N fixed magnetic field coils. Thereby, the aim is to control the plasma in such a way that its distribution function matches a desired distribution function at some certain final time T as closely as possible. This problem will be analyzed with respect to the following topics: existence of a globally optimal solution, necessary conditions of first order for local optimality, derivation of an optimality system, sufficient conditions of second order for local optimality and uniqueness of the optimal control under certain conditions. PubDate: 2018-09-25 DOI: 10.1007/s00245-018-9526-5

Authors:Boualem Djehiche; Said Hamadène Abstract: We establish existence of nearly-optimal controls, conditions for existence of an optimal control and a saddle-point for respectively a control problem and zero-sum differential game associated with payoff functionals of mean-field type, under dynamics driven by weak solutions of stochastic differential equations of mean-field type. PubDate: 2018-09-25 DOI: 10.1007/s00245-018-9525-6

Authors:Brahim El Asri; Sehail Mazid Abstract: This paper considers the problem of two-player zero-sum stochastic differential game with both players adopting impulse controls in finite horizon under rather weak assumptions on the cost functions (c and \(\chi \) not decreasing in time). We use the dynamic programming principle and viscosity solutions approach to show existence and uniqueness of a solution for the Hamilton–Jacobi–Bellman–Isaacs (HJBI) partial differential equation (PDE) of the game. We prove that the upper and lower value functions coincide. PubDate: 2018-09-24 DOI: 10.1007/s00245-018-9529-2

Authors:Sergio Frigeri; Maurizio Grasselli; Jürgen Sprekels Abstract: In this paper, we consider a two-dimensional diffuse interface model for the phase separation of an incompressible and isothermal binary fluid mixture with matched densities. This model consists of the Navier–Stokes equations, nonlinearly coupled with a convective nonlocal Cahn–Hilliard equation. The system rules the evolution of the (volume-averaged) velocity \(\varvec{u}\) of the mixture and the (relative) concentration difference \(\varphi \) of the two phases. The aim of this work is to study an optimal control problem for such a system, the control being a time-dependent external force \(\varvec{v}\) acting on the fluid. We first prove the existence of an optimal control for a given tracking type cost functional. Then we study the differentiability properties of the control-to-state map \(\varvec{v}\mapsto [\varvec{u},\varphi ]\) , and we establish first-order necessary optimality conditions. These results generalize the ones obtained by the first and the third authors jointly with Rocca (SIAM J Control Optim 54:221–250, 2016). There the authors assumed a constant mobility and a regular potential with polynomially controlled growth. Here, we analyze the physically more relevant case of a degenerate mobility and a singular (e.g., logarithmic) potential. This is made possible by the existence of a unique strong solution which was recently proved by the authors and Gal (WIAS preprint series No. 2309, Berlin, 2016). PubDate: 2018-09-24 DOI: 10.1007/s00245-018-9524-7

Authors:Maxime Laborde Abstract: In this paper, we investigate the existence of solution for systems of Fokker–Planck equations coupled through a common nonlinear congestion. Two different kinds of congestion are considered: a porous media congestion or soft congestion and the hard congestion given by the constraint \(\rho _1+\rho _2 \leqslant 1\) . We show that these systems can be seen as gradient flows in a Wasserstein product space and then we obtain a constructive method to prove the existence of solutions. Therefore it is natural to apply it for numerical purposes and some numerical simulations are included. PubDate: 2018-09-22 DOI: 10.1007/s00245-018-9527-4

Authors:Isaías Pereira de Jesus Abstract: The original version of this article unfortunately contained a mistake in the equation. PubDate: 2018-09-18 DOI: 10.1007/s00245-018-9522-9

Authors:Yang Shen; Jiaqin Wei; Qian Zhao Abstract: In this paper, we study an asset–liability management problem under a mean–variance criterion with regime switching. Unlike previous works, the dynamics of assets and liability are described by non-Markovian regime-switching models in the sense that all the model parameters are predictable with respect to the filtration generated jointly by a Markov chain and a Brownian motion. The problem is solved with the aid of backward stochastic differential equations (BSDEs) and bounded mean oscillation martingales. An efficient strategy and an efficient frontier are obtained and represented by unique solutions to several relevant BSDEs. We show that the optimal capital structure can be achieved when the initial asset value is expressed by a linear combination of the initial liability and the expected surplus. It is further found that a mutual fund theorem holds not only for the efficient strategy, but also for the optimal capital structure. PubDate: 2018-09-11 DOI: 10.1007/s00245-018-9523-8

Authors:Nikolaos S. Papageorgiou; Vicenţiu D. Rădulescu; Dušan D. Repovš Abstract: We consider a nonlinear elliptic equation driven by a nonhomogeneous differential operator plus an indefinite potential. On the reaction term we impose conditions only near zero. Using variational methods, together with truncation and perturbation techniques and critical groups, we produce three nontrivial solutions with sign information. In the semilinear case we improve this result by obtaining a second nodal solution for a total of four nontrivial solutions. Finally, under a symmetry condition on the reaction term, we generate a whole sequence of distinct nodal solutions. PubDate: 2018-09-07 DOI: 10.1007/s00245-018-9521-x

Authors:Joël Blot Abstract: We use an extension to the infinite dimension of the rank theorem of the differential calculus to establish a Lagrange theorem for optimization problems in Banach spaces. We provide an application to variational problems on a space of bounded sequences under equality constraints. PubDate: 2018-09-05 DOI: 10.1007/s00245-018-9520-y