Authors:Erhan Bayraktar; Zhou Zhou Pages: 457 - 468 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: 2018-12-01 DOI: 10.1007/s00245-017-9412-6 Issue No:Vol. 78, No. 3 (2018)

Authors:Géraldine Bouveret; Jean-François Chassagneux Pages: 469 - 491 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: 2018-12-01 DOI: 10.1007/s00245-017-9413-5 Issue No:Vol. 78, No. 3 (2018)

Authors:Harald Garcke; Kei Fong Lam; Elisabetta Rocca Pages: 495 - 544 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: 2018-12-01 DOI: 10.1007/s00245-017-9414-4 Issue No:Vol. 78, No. 3 (2018)

Authors:Michele Colturato Pages: 545 - 585 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: 2018-12-01 DOI: 10.1007/s00245-017-9415-3 Issue No:Vol. 78, No. 3 (2018)

Authors:O. L. V. Costa; F. Dufour Pages: 587 - 611 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: 2018-12-01 DOI: 10.1007/s00245-017-9416-2 Issue No:Vol. 78, No. 3 (2018)

Authors:Xiaopeng Zhao; Kung Fu Ng; Chong Li; Jen-Chih Yao Pages: 613 - 641 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: 2018-12-01 DOI: 10.1007/s00245-017-9417-1 Issue No:Vol. 78, No. 3 (2018)

Authors:Dirk Becherer; Todor Bilarev; Peter Frentrup Pages: 643 - 676 Abstract: We study a multiplicative transient price impact model for an illiquid financial market, where trading causes price impact which is multiplicative in relation to the current price, transient over time with finite rate of resilience, and non-linear in the order size. We construct explicit solutions for the optimal control and the value function of singular optimal control problems to maximize expected discounted proceeds from liquidating a given asset position. A free boundary problem, describing the optimal control, is solved for two variants of the problem where admissible controls are monotone or of bounded variation. PubDate: 2018-12-01 DOI: 10.1007/s00245-017-9418-0 Issue No:Vol. 78, No. 3 (2018)

Authors:Damien Lamberton Abstract: We consider the binomial approximation of the American put price in the Black–Scholes model (with continuous dividend yield). Our main result is that the error of approximation is \(O((\ln n )^\alpha /n)\) , where n is the number of time periods and the exponent \(\alpha \) is a positive number, the value of which may differ according to the respective levels of the interest rate and the dividend yield. PubDate: 2018-12-13 DOI: 10.1007/s00245-018-9545-2

Authors:Xiaoli Zhang; Huilai Li; Changchun Liu Abstract: In this paper, we consider a distributed optimal control problem for the Cahn–Hilliard/Allen–Cahn equation with state-constraint. The objective is to force the coverage y to have some specified properties or achieve a certain goal. Since the cost functional is discontinuous, together with state constraint, we employ a new penalty functional by the approximation of the cost functional, in this case, we derive the necessary optimality conditions for the approximating optimal control problem. Finally, by considering the limits of the necessary optimality conditions we have obtained, we solve the optimal control problem and derive the necessary optimality conditions. PubDate: 2018-12-12 DOI: 10.1007/s00245-018-9546-1

Authors:B. Feng; M. A. Jorge Silva; A. H. Caixeta Abstract: This is a complementation work of the paper referred in Jorge Silva, Muñoz Rivera and Racke (Appl Math Optim 73:165–194, 2016) where the authors proposed a semi-linear viscoelastic Kirchhoff plate model. While in [28] it is presented a study on well-posedness and energy decay rates in a historyless memory context, here our main goal is to consider the problem in a past history framework and then analyze its long-time behavior through the corresponding autonomous dynamical system. More specifically, our results are concerned with the existence of finite dimensional attractors as well as their intrinsic properties from the dynamical systems viewpoint. In addition, we also present a physical justification of the model under consideration. Hence, our new achievements complement those established in [28] to the case of memory in a history space setting and extend the results in Jorge Silva and Ma (IMA J Appl Math 78:1130–1146, 2013, J Math Phys 54:021505, 2013) to the case of dissipation only given by the memory term. PubDate: 2018-12-11 DOI: 10.1007/s00245-018-9544-3

Authors:Caijing Jiang; Biao Zeng Abstract: The paper investigates control problems for a class of nonlinear elliptic variational–hemivariational inequalities with constraint sets. Based on the well posedness of a variational–hemivariational inequality, we prove some results on continuous dependence and existence of optimal pairs to optimal control problems. We obtain some continuous dependence results in which the strong dependence and weak dependence are considered, respectively. A frictional contact problem is given to illustrate our main results. PubDate: 2018-12-05 DOI: 10.1007/s00245-018-9543-4

Authors:Tuomo Valkonen Abstract: Employing the ideas of non-linear preconditioning and testing of the classical proximal point method, we formalise common arguments in convergence rate and convergence proofs of optimisation methods to the verification of a simple iteration-wise inequality. When applied to fixed point operators, the latter can be seen as a generalisation of firm non-expansivity or the \(\alpha \) -averaged property. The main purpose of this work is to provide the abstract background theory for our companion paper “Block-proximal methods with spatially adapted acceleration”. In the present account we demonstrate the effectiveness of the general approach on several classical algorithms, as well as their stochastic variants. Besides, of course, the proximal point method, these method include the gradient descent, forward–backward splitting, Douglas–Rachford splitting, Newton’s method, as well as several methods for saddle-point problems, such as the Alternating Directions Method of Multipliers, and the Chambolle–Pock method. PubDate: 2018-11-28 DOI: 10.1007/s00245-018-9541-6

Authors:Karthik Adimurthi; Tadele Mengesha; Nguyen Cong Phuc Abstract: Local and global weighted norm estimates involving Muckenhoupt weights are obtained for gradient of solutions to linear elliptic Dirichlet boundary value problems in divergence form over a Lipschitz domain \(\Omega \) . The gradient estimates are obtained in weighted Lebesgue and Lorentz spaces, which also yield estimates in Lorentz–Morrey spaces as well as Hölder continuity of solutions. The significance of the work lies on its applicability to very weak solutions (that belong to \(W^{1,p}_{0}(\Omega )\) for some \(p>1\) but not necessarily in \(W^{1,2}_{0}(\Omega )\) ) to inhomogeneous equations with coefficients that may have discontinuities but have a small mean oscillation. The domain is assumed to have a Lipschitz boundary with small Lipschitz constant and as such allows corners. The approach implemented makes use of localized sharp maximal function estimates as well as known regularity estimates for very weak solutions to the associated homogeneous equations. The estimates are optimal in the sense that they coincide with classical weighted gradient estimates in the event the coefficients are continuous and the domain has smooth boundary. PubDate: 2018-11-28 DOI: 10.1007/s00245-018-9542-5

Authors:Pierluigi Colli; Gianni Gilardi; Jürgen Sprekels Abstract: In the recent paper “Well-posedness and regularity for a generalized fractional Cahn–Hilliard system” by the same authors, general well-posedness results have been established for a class of evolutionary systems of two equations having the structure of a viscous Cahn–Hilliard system, in which nonlinearities of double-well type occur. The operators appearing in the system equations are fractional versions in the spectral sense of general linear operators A, B having compact resolvents, which are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. In this work we complement the results given in the quoted paper by studying a distributed control problem for this evolutionary system. The main difficulty in the analysis is to establish a rigorous Fréchet differentiability result for the associated control-to-state mapping. This seems only to be possible if the state stays bounded, which, in turn, makes it necessary to postulate an additional global boundedness assumption. One typical situation, in which this assumption is satisfied, arises when B is the negative Laplacian with zero Dirichlet boundary conditions and the nonlinearity is smooth with polynomial growth of at most order four. Also a case with logarithmic nonlinearity can be handled. Under the global boundedness assumption, we establish existence and first-order necessary optimality conditions for the optimal control problem in terms of a variational inequality and the associated adjoint state system. PubDate: 2018-11-15 DOI: 10.1007/s00245-018-9540-7

Authors:János Flesch; Arkadi Predtetchinski; William Sudderth Abstract: A positive zero-sum stochastic game with countable state and action spaces is shown to have a value if, at every state, at least one player has a finite action space. The proof uses transfinite algorithms to calculate the upper and lower values of the game. We also investigate the existence of ( \(\epsilon \) -)optimal strategies in the classes of stationary and Markov strategies. PubDate: 2018-11-01 DOI: 10.1007/s00245-018-9536-3

Authors:Xin-Guang Yang; Jing Zhang; Yongjin Lu Abstract: This paper is concerned with the wellposedness of global solution and existence of global attractor to the nonlinear Timoshenko system subject to continuous variable time delay in the angular rotation of the beam filament. The waves are assumed to propagate under the same speed in the transversal and angular direction. A single mechanical damping is implemented to counter the destabilizing effect from the time delay term. By imposing appropriate assumptions on the damping term and sub-linear time delay term, we prove the existence of absorbing set and establish the quasi-stability of the gradient system generated from the solution to the system of equation. The quasi-stability property in turn implies the existence of finite dimensional global and exponential attractors that contain the unstable manifold formed from the set of equilibria. PubDate: 2018-11-01 DOI: 10.1007/s00245-018-9539-0

Authors:Shitao Liu; Yang Yang Abstract: We study an inverse boundary value problem with partial data in an infinite slab in \(\mathbb {R}^{n}\) , \(n\ge 3\) , for the magnetic Schrödinger operator with bounded magnetic potential and electric potential. We show that the magnetic field and the electric potential can be uniquely determined, when the Dirichlet and Neumann data are given on either different boundary hyperplanes or on the same boundary hyperplanes of the slab. These generalize the results in Krupchyk et al. (Commun Math Phys 312:87–126, 2012), where the same uniqueness results were established when the magnetic potential is Lipschitz continuous. The proof is based on the complex geometric optics solutions constructed in Krupchyk and Uhlmann (Commun Math Phys 327:993–1009, 2014), which are special solutions to the magnetic Schrödinger equation with \(L^{\infty }\) magnetic and electric potentials in a bounded domain. PubDate: 2018-10-30 DOI: 10.1007/s00245-018-9537-2

Authors:Weiwei Hu Abstract: We consider an approximating control design for optimal mixing of a non-dissipative scalar field \(\theta \) in an unsteady Stokes flow. The objective of our approach is to achieve optimal mixing at a given final time \(T>0\) , via the active control of the flow velocity v through boundary inputs. Due to zero diffusivity of the scalar field \(\theta \) , establishing the well-posedness of its Gâteaux derivative requires \(\sup _{t\in [0,T]}\Vert \nabla \theta \Vert _{L^2}<\infty \) , which in turn demands the flow velocity field to satisfy the condition \(\int ^{T}_{0}\Vert \nabla v\Vert _{L^{\infty }(\Omega )}\, dt<\infty \) . This condition results in the need to penalize the time derivative of the boundary control in the cost functional. Consequently, the optimality system becomes difficult to solve (Hu in Appl Math Optim 78(1):201–217, 2018). Our current approximating approach provides a more transparent optimality system, with the set of admissible controls square integrable in space-time. This is achieved by first introducing a small diffusivity to the scalar equation and then establishing a rigorous analysis of convergence of the approximating control problem to the original one as the diffusivity approaches to zero. Uniqueness of the optimal solution is obtained for the two dimensional case. PubDate: 2018-10-30 DOI: 10.1007/s00245-018-9535-4

Authors:Andrea Signori Abstract: This paper is intended to tackle the control problem associated with an extended phase field system of Cahn–Hilliard type that is related to a tumor growth model. This system has been investigated in previous contributions from the viewpoint of well-posedness and asymptotic analyses. Here, we aim to extend the mathematical studies around this system by introducing a control variable and handling the corresponding control problem. We try to keep the potential as general as possible, focusing our investigation towards singular potentials, such as the logarithmic one. We establish the existence of optimal control, the Lipschitz continuity of the control-to-state mapping and even its Fréchet differentiability in suitable Banach spaces. Moreover, we derive the first-order necessary conditions that an optimal control has to satisfy. PubDate: 2018-10-30 DOI: 10.1007/s00245-018-9538-1