Authors:Sérgio S. Rodrigues Abstract: Abstract Given a nonstationary trajectory of the Navier–Stokes system, a finite-dimensional feedback boundary control stabilizing locally the system to the given trajectory is derived. Moreover the control is supported in a given open subset of the boundary of the domain containing the fluid. In a first step a controller (feedback operator) is derived which stabilizes the linear Oseen–Stokes system “around the given trajectory” to zero; for that a corollary of a suitable truncated boundary observability inequality, the regularizing property for the system, and some standard techniques of the optimal control theory are used. Then it is shown that the same controller also stabilizes, locally, the Navier–Stokes system to the given trajectory. PubDate: 2018-01-06 DOI: 10.1007/s00245-017-9474-5

Authors:Ana Cristina Barroso; Elvira Zappale Abstract: Abstract In this paper we investigate the possibility of obtaining a measure representation for functionals arising in the context of optimal design problems under non-standard growth conditions and perimeter penalization. Applications to modelling of strings are also provided. PubDate: 2018-01-05 DOI: 10.1007/s00245-017-9473-6

Authors:Ugur G. Abdulla; Bruno Poggi Abstract: Abstract We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria consist of the minimization of the \(L_2\) -norm declination of the trace of the solution to the Stefan problem from the temperature measurement on the fixed right boundary. The state vector solves multiphase Stefan problem in a weak formulation, which is equivalent to Neumann problem for the quasilinear parabolic PDE with discontinuous coefficient. Full discretization through finite differences is implemented and discrete optimal control problem is introduced. We prove well-posedness in a Sobolev space framework and convergence of discrete optimal control problems to the original problem both with respect to the cost functional and control. Along the way, the convergence of the method of finite differences for the weak solution of the multiphase Stefan problem is proved. The proof is based on achieving a uniform \(L_{\infty }\) bound, and \(W_2^{1,1}\) -energy estimate for the discrete multiphase Stefan problem. PubDate: 2018-01-03 DOI: 10.1007/s00245-017-9472-7

Authors:D. T. V. An; J.-C. Yao; N. D. Yen Abstract: Abstract A parametric constrained convex optimal control problem, where the initial state is perturbed and the linear state equation contains a noise, is considered in this paper. Formulas for computing the subdifferential and the singular subdifferential of the optimal value function at a given parameter are obtained by means of some recent results on differential stability in mathematical programming. The computation procedures and illustrative examples are presented. PubDate: 2018-01-03 DOI: 10.1007/s00245-017-9475-4

Authors:Radouen Ghanem; Ibtissam Nouri Pages: 465 - 500 Abstract: Abstract We consider an optimal control problem for the obstacle problem with an elliptic polyharmonic obstacle problem of order 2m, where the obstacle function is assumed to be the control. We use a Moreau–Yosida approximate technique to introduce a family of problems governed by variational equations. Then, we prove optimal solutions existence and give an approximate optimality system and convergence results by passing to the limit in this system. PubDate: 2017-12-01 DOI: 10.1007/s00245-016-9358-0 Issue No:Vol. 76, No. 3 (2017)

Authors:Guillaume Carlier; Xavier Dupuis Pages: 565 - 592 Abstract: Abstract The principal-agent problem in economics leads to variational problems subject to global constraints of b-convexity on the admissible functions, capturing the so-called incentive-compatibility constraints. Typical examples are minimization problems subject to a convexity constraint. In a recent pathbreaking article, Figalli et al. (J Econ Theory 146(2):454–478, 2011) identified conditions which ensure convexity of the principal-agent problem and thus raised hope on the development of numerical methods. We consider special instances of projections problems over b-convex functions and show how they can be solved numerically using Dykstra’s iterated projection algorithm to handle the b-convexity constraint in the framework of (Figalli et al. in J Econ Theory 146(2):454–478, 2011). Our method also turns out to be simple for convex envelope computations. PubDate: 2017-12-01 DOI: 10.1007/s00245-016-9361-5 Issue No:Vol. 76, No. 3 (2017)

Authors:P. Mondal; S. K. Neogy; S. Sinha; D. Ghorui Pages: 593 - 619 Abstract: Abstract In this article, we revisit the applications of principal pivot transform and its generalization for a rectangular matrix (in the context of vertical linear complementarity problem) to solve some structured classes of zero-sum two-person discounted semi-Markov games with finitely many states and actions. The single controller semi-Markov games have been formulated as a linear complementarity problem and solved using a stepwise principal pivoting algorithm. We provide a sufficient condition for such games to be completely mixed. The concept of switching controller semi-Markov games is introduced and we prove the ordered field property and the existence of stationary optimal strategies for such games. Moreover, such games are formulated as a vertical linear complementarity problem and have been solved using a stepwise generalized principal pivoting algorithm. Sufficient conditions are also given for such games to be completely mixed. For both these classes of games, some properties analogous to completely mixed matrix games, are established. PubDate: 2017-12-01 DOI: 10.1007/s00245-016-9362-4 Issue No:Vol. 76, No. 3 (2017)

Authors:Filippo Dell’Oro; Vittorino Pata Pages: 641 - 655 Abstract: Abstract We discuss the parallel between the third-order Moore–Gibson–Thompson equation $$\begin{aligned} {\partial _{ttt}} u + \alpha {\partial _{tt}}u-\beta \Delta {\partial _t} u - \gamma \Delta u =0 \end{aligned}$$ depending on the parameters \(\alpha ,\beta ,\gamma >0,\) and the equation of linear viscoelasticity $$\begin{aligned} \partial _{tt}u(t) - \kappa (0)\Delta u(t) - \int _{0}^\infty \kappa ^{\prime }(s)\Delta u(t-s)\,\mathrm{d}s=0 \end{aligned}$$ for the particular choice of the exponential kernel $$\begin{aligned} \kappa (s) = a \mathrm{e}^{-b s} + c \end{aligned}$$ with \(a,b,c>0\) . In particular, the latter model is shown to exhibit a preservation of regularity for a certain class of initial data, which is unexpected in presence of a general memory kernel \(\kappa \) . PubDate: 2017-12-01 DOI: 10.1007/s00245-016-9365-1 Issue No:Vol. 76, No. 3 (2017)

Authors:M. Pellicer; B. Said-Houari Abstract: Abstract In this paper, we study the Moore–Gibson–Thompson equation in \(\mathbb {R}^N\) , which is a third order in time equation that arises in viscous thermally relaxing fluids and also in viscoelastic materials (then under the name of standard linear viscoelastic model). First, we use some Lyapunov functionals in the Fourier space to show that, under certain assumptions on some parameters in the equation, a norm related to the solution decays with a rate \((1+t)^{-N/4}\) . Since the decay of the previous norm does not give the decay rate of the solution itself then, in the second part of the paper, we show an explicit representation of the solution in the frequency domain by analyzing the eigenvalues of the Fourier image of the solution and writing the solution accordingly. We use this eigenvalues expansion method to give the decay rate of the solution (and also of its derivatives), which results in \((1+t)^{1-N/4}\) for \(N=1,2\) and \((1+t)^{1/2-N/4}\) when \(N\ge 3\) . PubDate: 2017-12-30 DOI: 10.1007/s00245-017-9471-8

Authors:To Fu Ma; Rodrigo Nunes Monteiro; Ana Claudia Pereira Abstract: Abstract This paper is concerned with the Timoshenko system, a recognized model for vibrations of thin prismatic beams. The corresponding autonomous system has been widely studied. However, there are only a few works dedicated to its non-autonomous counterpart. Here, we investigate the long-time dynamics of Timoshenko systems involving a nonlinear foundation and subjected to perturbations of time-dependent external forces. The main result establishes the existence of a pullback exponential attractor, which as a consequence, implies the existence of a minimal pullback attractor with finite fractal dimension. The upper-semicontinuity of attractors, as the non-autonomous forces tend to zero, is also studied. PubDate: 2017-12-28 DOI: 10.1007/s00245-017-9469-2

Authors:Sandra Cerrai; Arnaud Debussche Abstract: Abstract We are dealing with the validity of a large deviation principle for a class of reaction–diffusion equations with polynomial non-linearity, perturbed by a Gaussian random forcing. We are here interested in the regime where both the strength of the noise and its correlation are vanishing, on a length scale \(\epsilon \) and \(\delta (\epsilon )\) , respectively, with \(0<\epsilon ,\delta (\epsilon )\ll 1\) . We prove that, under the assumption that \(\epsilon \) and \(\delta (\epsilon )\) satisfy a suitable scaling limit, a large deviation principle holds in the space of continuous trajectories with values both in the space of square-integrable functions and in Sobolev spaces of negative exponent. Our result is valid, without any restriction on the degree of the polynomial nor on the space dimension. PubDate: 2017-12-26 DOI: 10.1007/s00245-017-9459-4

Authors:Xiaoshan Chen; Zhen-Qing Chen; Ky Tran; George Yin Abstract: Abstract This work develops asymptotic properties of a class of switching jump diffusion processes. The processes under consideration may be viewed as a number of jump diffusion processes modulated by a random switching mechanism. The underlying processes feature in the switching process depends on the jump diffusions. In this paper, conditions for recurrence and positive recurrence are derived. Ergodicity is examined in detail. Existence of invariant probability measures is proved. PubDate: 2017-12-26 DOI: 10.1007/s00245-017-9470-9

Authors:T. E. Duncan; B. Maslowski; B. Pasik-Duncan Abstract: Abstract A stochastic linear-quadratic control problem is formulated and solved for some stochastic equations in an infinite dimensional Hilbert space for both finite and infinite time horizons. The equations are bilinear in the state and the noise process where the noise is a scalar Gauss-Volterra process. The Gauss-Volterra noise processes are obtained from the integral of a Brownian motion with a suitable kernel function. These noise processes include fractional Brownian motions with the Hurst parameter \(H \in (\frac{1}{2},1)\) , Liouville fractional Brownian motions with \(H \in (\frac{1}{2},1)\) , and some multifractional Brownian motions. The family of admissible controls for the quadratic costs is a family of linear feedback controls. This restriction on the family of controls allows for a feasible implementation of the optimal controls. The bilinear equations have drift terms that are linear evolution operators. These equations can model stochastic partial differential equations of parabolic and hyperbolic types and two families of examples are given. PubDate: 2017-12-22 DOI: 10.1007/s00245-017-9468-3

Authors:Pablo Azcue; Nora Muler Abstract: Abstract This paper presents a continuous time cash management problem where the uncontrolled money stock follows a compound poisson process with two-sided jumps and negative drift. The main goal is to minimize the total expected discounted sum of the opportunity cost of holding cash and the adjustment cost coming from deposits and withdrawals. This adjustment cost can have a proportional and a fixed component. In this paper it is assumed that the controlled money stock cannot be negative. We show that the optimal value function is an a.e. strong solution of the corresponding Hamilton–Jacobi–Bellman equation and that it can be characterized as the largest a.e. strong supersolution. We prove that the optimal policy has an impulse multi-band structure with multiple trigger-target pairs. We present examples where the optimal value function is not differentiable and an example where the optimal impulse policy has more that one band. PubDate: 2017-12-21 DOI: 10.1007/s00245-017-9467-4

Authors:Nikolaos S. Papageorgiou; Vicenţiu D. Rădulescu; Dušan D. Repovš Abstract: Abstract We consider a nonlinear boundary value problem driven by a nonhomogeneous differential operator. The problem exhibits competing nonlinearities with a superlinear (convex) contribution coming from the reaction term and a sublinear (concave) contribution coming from the parametric boundary (source) term. We show that for all small parameter values \(\lambda >0\) , the problem has at least five nontrivial smooth solutions, four of constant sign and one nodal. We also produce extremal constant sign solutions and determine their monotonicity and continuity properties as the parameter \(\lambda >0\) varies. In the semilinear case we produce a sixth nontrivial solution but without any sign information. Our approach uses variational methods together with truncation and perturbation techniques, and Morse theory. PubDate: 2017-12-13 DOI: 10.1007/s00245-017-9465-6

Authors:Exequiel Mallea-Zepeda; Elva Ortega-Torres; Élder J. Villamizar-Roa Abstract: Abstract We study an optimal boundary control problem for the two-dimensional stationary micropolar fluids system with variable density. We control the system by considering boundary controls, for the velocity vector and angular velocity of rotation of particles, on parts of the boundary of the flow domain. On the remaining part of the boundary, we consider mixed boundary conditions for the vector velocity (Dirichlet and Navier conditions) and Dirichlet boundary conditions for the angular velocity. We analyze the existence of a weak solution obtaining the fluid density as a scalar function in terms of the stream function. We prove the existence of an optimal solution and, by using the Lagrange multipliers theorem, we state first-order optimality conditions. We also derive, through a penalty method, some optimality conditions satisfied by the optimal controls. PubDate: 2017-12-13 DOI: 10.1007/s00245-017-9466-5

Authors:Xun Li; Jingrui Sun; Jie Xiong Abstract: Abstract This paper is concerned with linear quadratic optimal control problems for mean-field backward stochastic differential equations (MF-BSDEs, for short) with deterministic coefficients. The optimality system, which is a linear mean-field forward–backward stochastic differential equation with constraint, is obtained by a variational method. By decoupling the optimality system, two coupled Riccati equations and an MF-BSDE are derived. It turns out that the coupled two Riccati equations are uniquely solvable. Then a complete and explicit representation is obtained for the optimal control. PubDate: 2017-12-07 DOI: 10.1007/s00245-017-9464-7

Authors:Anatolii A. Puhalskii Abstract: Abstract We study the problem of optimal long term investment with a view to beat a benchmark for a diffusion model of asset prices. Two kinds of objectives are considered. One criterion concerns the probability of outperforming the benchmark and seeks either to minimise the decay rate of the probability that a portfolio exceeds the benchmark or to maximise the decay rate that the portfolio falls short. The other criterion concerns the growth rate of an expected risk-sensitised utility of wealth which has to be either minimised, for a risk-averse investor, or maximised, for a risk-seeking investor. It is assumed that the mean returns and volatilities of the securities are affected by an economic factor, possibly, in a nonlinear fashion. The economic factor and the benchmark are modelled with general Itô differential equations. The results identify asymptotically optimal portfolios and produce the decay, or growth, rates. The proportions of wealth invested in the individual securities are time-homogeneous functions of the economic factor. Furthermore, a uniform treatment is given to the out—and under—performance probability optimisation as well as to the risk-averse and risk-seeking portfolio optimisation. It is shown that there exists a portfolio that optimises the decay rates of both the outperformance probability and the underperformance probability. While earlier research on the subject has relied, for the most part, on the techniques of stochastic optimal control and dynamic programming, in this contribution the quantities of interest are studied directly by employing the methods of the large deviation theory. The key to the analysis is to recognise the setup in question as a case of coupled diffusions with time scale separation, with the economic factor representing “the fast motion”. PubDate: 2017-12-04 DOI: 10.1007/s00245-017-9457-6

Authors:Jean-Pierre Vila; Jean-Pierre Gauchi Abstract: Abstract This paper presents the main principles of a stochastic nonlinear model predictive control (NMPC) novel approach for imperfectly observed discrete time systems described by time varying nonlinear non-Gaussian state space models with unknown parameters. A convergent particle estimator of the conditional expectation of a chosen cost function on a given receding control horizon is built, leading to an almost sure (a.s.) epi-convergent estimator of the NMPC cost-to-go criterion. The estimator of the expected cost function relies upon simulations and on a recently developed nonparametric convergent particle estimator of a multi-step ahead conditional probability density function (pdf) of the state variables. The theory of stochastic epi-convergence is applied to the estimated cost-to-go criterion to prove the almost sure convergence of the optimal solutions of the approximated NMPC problem to their true counterparts, when both simulation and particle numbers grow to infinity. PubDate: 2017-12-01 DOI: 10.1007/s00245-017-9462-9

Authors:Erhan Bayraktar; Jiaqi Li Abstract: Abstract We analyze the continuous time zero-sum and cooperative controller-stopper games of Karatzas and Sudderth (Ann Probab 29(3):1111–1127, 2001), Karatzas and Zamfirescu (Ann Probab 36(4):1495–1527, 2008) and Karatzas and Zamfirescu (Appl Math Optim 53(2):163–184, 2006) when the volatility of the state process is controlled as in Bayraktar and Huang (SIAM J Control Optim 51(2):1263–1297, 2013) but additionally when the state process has controlled jumps. We perform this analysis by first resolving the stochastic target problems [of Soner and Touzi (SIAM J Control Optim 41(2):404–424, 2002; J Eur Math Soc 4(3):201–236, 2002)] with a cooperative or a non-cooperative stopper and then embedding the original problem into the latter set-up. Unlike in Bayraktar and Huang (SIAM J Control Optim 51(2):1263–1297, 2013) our analysis relies crucially on the Stochastic Perron method of Bayraktar and Sîrbu (SIAM J Control Optim 51(6):4274–4294, 2013) but not the dynamic programming principle, which is difficult to prove directly for games. PubDate: 2017-12-01 DOI: 10.1007/s00245-017-9463-8