Authors:Abdelkarim Kelleche; Nasser-eddine Tatar Pages: 343 - 364 Abstract: The paper deals with an axially moving viscoelastic structure modeled as an Euler–Bernoulli beam. The aim is to suppress the transversal displacement (transversal vibrations) that occur during the axial motion of the beam. It is assumed that the beam is moving with a constant axial speed and it is subject to a nonlinear force at the right boundary. We prove that when the axial speed of the beam is smaller than a critical value, the dissipation produced by the viscoelastic material is sufficient to suppress the transversal vibrations. It is shown that the rate of decay of the energy depends on the kernel which arise in the viscoelastic term. We consider a general kernel and notice that solutions cannot decay faster than the kernel. PubDate: 2017-06-01 DOI: 10.1007/s00245-016-9334-8 Issue No:Vol. 75, No. 3 (2017)

Authors:I McGillivray Pages: 365 - 401 Abstract: Let U be a bounded open connected set in \(\mathbb {R}^n\) ( \(n\ge 1\) ). We refer to the unique weak solution of the Poisson problem \(-\Delta u = \chi _A\) on U with Dirichlet boundary conditions as \(u_A\) for any measurable set A in U. The function \(\psi :=u_U\) is the torsion function of U. Let V be the measure \(V:=\psi \,\mathscr {L}^n\) on U where \(\mathscr {L}^n\) stands for n-dimensional Lebesgue measure. We study the variational problem $$\begin{aligned} I(U,p):=\sup \Big \{ J(A)-V(U)\,p^2\,\Big \} \end{aligned}$$ with \(p\in (0,1)\) where \(J(A):=\int _Au_A\,dx\) and the supremum is taken over measurable sets \(A\subset U\) subject to the constraint \(V(A)=pV(U)\) . We relate the above problem to an unstable two-phase membrane problem. We characterise optimsers in the case \(n=1\) . The proof makes use of weighted isoperimetric and Pólya–Szegö inequalities. PubDate: 2017-06-01 DOI: 10.1007/s00245-016-9335-7 Issue No:Vol. 75, No. 3 (2017)

Authors:Maisa Khader; Belkacem Said-Houari Pages: 403 - 428 Abstract: We consider the Cauchy problem for the one-dimensional Timoshenko system coupled with the heat conduction, wherein the latter is described by the Gurtin–Pipkin thermal law. We study the decay properties of the system using the energy method in the Fourier space (to build an appropriate Lyapunov functional) accompanied with some integral estimates. We show that the number \(\alpha _g\) (depending on the parameters of the system) found in (Dell’Oro and Pata J Differ Equ 257(2):523–548, 2014), which rules the evolution in bounded domains, also plays a role in an unbounded domain and controls the behavior of the solution. In fact, we prove that if \(\alpha _g=0,\) then the \(L^2\) -norm of the solution decays with the rate \((1+t)^{-1/12}\) . The same decay rate has been obtained for \(\alpha _g\ne 0,\) but under some higher regularity assumption. This high regularity requirement is known as regularity loss, which means that in order to get the estimate for the \(H^s\) -norm of the solution, we need our initial data to be in the space \(H^{s+s_0},\ s_0>1\) . PubDate: 2017-06-01 DOI: 10.1007/s00245-016-9336-6 Issue No:Vol. 75, No. 3 (2017)

Authors:Giorgio Ferrari; Frank Riedel; Jan-Henrik Steg Pages: 429 - 470 Abstract: In this paper we study continuous-time stochastic control problems with both monotone and classical controls motivated by the so-called public good contribution problem. That is the problem of n economic agents aiming to maximize their expected utility allocating initial wealth over a given time period between private consumption and irreversible contributions to increase the level of some public good. We investigate the corresponding social planner problem and the case of strategic interaction between the agents, i.e. the public good contribution game. We show existence and uniqueness of the social planner’s optimal policy, we characterize it by necessary and sufficient stochastic Kuhn–Tucker conditions and we provide its expression in terms of the unique optional solution of a stochastic backward equation. Similar stochastic first order conditions prove to be very useful for studying any Nash equilibria of the public good contribution game. In the symmetric case they allow us to prove (qualitative) uniqueness of the Nash equilibrium, which we again construct as the unique optional solution of a stochastic backward equation. We finally also provide a detailed analysis of the so-called free rider effect. PubDate: 2017-06-01 DOI: 10.1007/s00245-016-9337-5 Issue No:Vol. 75, No. 3 (2017)

Authors:Jianliang Zhai; Tusheng Zhang Pages: 471 - 498 Abstract: In this paper, we establish a large deviation principle for stochastic models of incompressible second grade fluids. The weak convergence method introduced by Budhiraja and Dupuis (Probab Math Statist 20:39–61, 2000) plays an important role. PubDate: 2017-06-01 DOI: 10.1007/s00245-016-9338-4 Issue No:Vol. 75, No. 3 (2017)

Authors:Tijana Levajković; Hermann Mena; Amjad Tuffaha Pages: 499 - 523 Abstract: We present an approximation framework for computing the solution of the stochastic linear quadratic control problem on Hilbert spaces. We focus on the finite horizon case and the related differential Riccati equations (DREs). Our approximation framework is concerned with the so-called “singular estimate control systems” (Lasiecka in Optimal control problems and Riccati equations for systems with unbounded controls and partially analytic generators: applications to boundary and point control problems, 2004) which model certain coupled systems of parabolic/hyperbolic mixed partial differential equations with boundary or point control. We prove that the solutions of the approximate finite-dimensional DREs converge to the solution of the infinite-dimensional DRE. In addition, we prove that the optimal state and control of the approximate finite-dimensional problem converge to the optimal state and control of the corresponding infinite-dimensional problem. PubDate: 2017-06-01 DOI: 10.1007/s00245-016-9339-3 Issue No:Vol. 75, No. 3 (2017)

Authors:Hessah Al Motairi; Mihail Zervos Pages: 525 - 551 Abstract: We consider an irreversible capacity expansion model in which additional investment has a strictly negative effect on the value of an underlying stochastic economic indicator. The associated optimisation problem takes the form of a singular stochastic control problem that admits an explicit solution. A special characteristic of this stochastic control problem is that changes of the state process due to control action depend on the state process itself in a proportional way. PubDate: 2017-06-01 DOI: 10.1007/s00245-016-9341-9 Issue No:Vol. 75, No. 3 (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:Joscha Diehl; Peter K. Friz; Paul Gassiat Pages: 285 - 315 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:Jing Liu; Song Yao Abstract: Given \(p \in (1,2]\) , the wellposedness of backward stochastic differential equations with jumps (BSDEJs) in \(\mathbb {L}^p\) sense gives rise to a so-called g-expectation with \(\mathbb {L}^p\) domain under the jump filtration (the one generated by a Brownian motion and a Poisson random measure). In this paper, we extend such a g-expectation to a nonlinear expectation \(\mathcal{E}\) with \(\mathbb {L}^p\) domain that is consistent with the jump filtration. We study the basic (martingale) properties of the jump-filtration consistent nonlinear expectation \(\mathcal{E}\) and show that under certain domination condition, the nonlinear expectation \(\mathcal{E}\) can be represented by some g-expectation. PubDate: 2017-05-20 DOI: 10.1007/s00245-017-9422-4

Authors:Fatima Zahra Mokkedem; Xianlong Fu Abstract: In this paper we study the standard optimal control and time optimal control problems for a class of semilinear evolution systems with infinite delay. We first establish the results of existence and uniqueness of mild solution and the compactness of the solution operator for the control system. Then, based on these results, we investigate the optimal control problem with integral cost function and the time optimal control problem respectively. Under some conditions we show the existence of optimal controls for the both cases of bounded and unbounded admissible control sets. We also obtain the existence of time optimal control to a target set. In addition, a convergence theorem of time optimal controls to a point target set is proved. Finally, an example is given to show the application of the main results. PubDate: 2017-05-13 DOI: 10.1007/s00245-017-9420-6

Authors:Kazuo Yamazaki Abstract: The theory of micropolar fluids emphasizes the micro-structure of fluids by coupling the Navier–Stokes equations with micro-rotational velocity, and is widely viewed to be well fit, better than the Navier–Stokes equations, to describe fluids consisting of bar-like elements such as liquid crystals made up of dumbbell molecules or animal blood. Following the work of Weinan et al. (Commun Math Phys 224:83–106, 2001), we prove the existence of a unique stationary measure for the stochastic micropolar fluid system with periodic boundary condition, forced by only the determining modes of the noise and therefore a type of finite-dimensionality of micropolar fluid flow. The novelty of the manuscript is a series of energy estimates that is reminiscent from analysis in the deterministic case. PubDate: 2017-05-10 DOI: 10.1007/s00245-017-9419-z

Authors:Annalisa Massaccesi; Edouard Oudet; Bozhidar Velichkov Abstract: In this paper we propose a variational approach to the Steiner tree problem, which is based on calibrations in a suitable algebraic environment for polyhedral chains which represent our candidates. This approach turns out to be very efficient from numerical point of view and allows to establish whether a given Steiner tree is optimal. Several examples are provided. PubDate: 2017-05-10 DOI: 10.1007/s00245-017-9421-5

Authors:Xiaopeng Zhao; Kung Fu Ng; Chong Li; Jen-Chih Yao Abstract: For a finite/infinite family of closed convex sets with nonempty intersection in Hilbert space, we consider the (bounded) linear regularity property and the linear convergence property of the projection-based methods for solving the convex feasibility problem. Several sufficient conditions are provided to ensure the bounded linear regularity in terms of the interior-point conditions and some finite codimension assumptions. A unified projection method, called Algorithm B-EMOPP, for solving the convex feasibility problem is proposed, and by using the bounded linear regularity, the linear convergence results for this method are established under a new control strategy introduced here. PubDate: 2017-05-02 DOI: 10.1007/s00245-017-9417-1

Authors:O. L. V. Costa; F. Dufour 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: 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: 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: 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: 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: 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