Maximum Principle of Optimal Control of the Primitive Equations of the Ocean With State Constraint

We investigate in this paper Pontryagin's maximum principle for a class of control problems associated with the primitive equations (PEs) of the ocean. These optimal problems involve a state constraint similar to that considered in Wang and Wang (Nonlinear Analysis 2003; 52:1911–1931) for the three-dimensional Navier–Stokes (NS) equations. The main difference between this work and Wang and Wang (Nonlinear Analysis 2003; 52:1911–1931) is that the nonlinearity in the PEs is stronger than in the three-dimensional NS systems.     

Second-order Optimality Conditions for Optimal Control of the Primitive Equations of the Ocean with Periodic Inputs

We investigate in this article the Pontryagin’s maximum principle for control problem associated with the primitive equations (PEs) of the ocean with periodic inputs. We also derive a second-order sufficient condition for optimality. This work is closely related to Wang (SIAM J. Control Optim. 41(2):583–606, 2002) and He (Acta Math. Sci. Ser. B Engl. Ed. 26(4):729–734, 2006), in which the authors proved similar results for the three-dimensional Navier-Stokes (NS) systems.     

Barotropic-Baroclinic Formulation of the Primitive Equations of the Ocean

In this article, we present an equivalent barotropic-baroclinic formulation of the primitive equations (PEs) of the ocean given in [J.L. Lions, R. Temam and S. Wang (<citeref rid="bib5">1992</citeref>). On the equations of large-scale ocean. Nonlinearity, 5, 1007-1053.]. From the numerical point of view, the main advantage of this new formulation is that the incompressibility condition appearing in the PEs in [J.L. Lions, R. Temam and S. Wang (<citeref rid="bib5">1992</citeref>). On the equations of large-scale ocean. Nonlinearity, 5, 1007-1053.] is automatically satisfied without being explicitly imposed at any stage. Some numerical schemes for the time integration of the PEs are presented and their numerical stability is discussed. These schemes are reminiscent of other schemes that have been used for other equations in particular the Navier-Stokes equations. We end the article by presenting numerical simulations of a wind-driven ocean model using the new formulation. More extensive numerical simulations and physical aspects will be presented elsewhere.     

New blow-up rates for fast controls of Schrödinger and heat equations

We consider the null-controllability problem for the Schrödinger and heat equations with boundary control. We concentrate on short-time, or fast, controls. We improve recent estimates (see [L. Miller, Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J. Differential Equations 204 (2004) 202-226; L. Miller, How violent are fast controls for Schrödinger and plate vibrations?, Arch. Ration. Mech. Anal. 172 (2004) 429-456; L. Miller, Controllability cost of conservative systems: Resolvent condition and transmutation, J. Funct. Anal. 218 (2005) 425-444; L. Miller, The control transmutation method and the cost of fast controls, SIAM J. Control Optim. 45 (2006) 762-772]) on the norm of the operator associating to any initial state the minimal norm control driving the system to zero. Our main results concern the Schrödinger and heat equations in one space dimension. They yield new estimates concerning window problems for series of exponentials as described in [T.I. Seidman, The coefficient map for certain exponential sums, Nederl. Akad. Wetensch. Indag. Math. 48 (1986) 463-478] and in [T.I. Seidman, S.A. Avdonin, S.A. Ivanov, The “window problem” for series of complex exponentials, J. Fourier Anal. Appl. 6 (2000) 233-254]. These results are used, following [L. Miller, The control transmutation method and the cost of fast controls, SIAM J. Control Optim. 45 (2006) 762-772], to deal with the case of several space dimensions.     

Asymptotic behavior of the Korteweg-de Vries equation posed in a quarter plane

The purpose of this work is to study the exponential stabilization of the Korteweg-de Vries equation in the right half-line under the effect of a localized damping term. We follow the methods in [G.P. Menzala, C.F. Vasconcellos, E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quart. Appl. Math. LX (1) (2002) 111-129] which combine multiplier techniques and compactness arguments and reduce the problem to prove the unique continuation property of weak solutions. Here, the unique continuation is obtained in two steps: we first prove that solutions vanishing on the support of the damping function are necessarily smooth and then we apply the unique continuation results proved in [J.C. Saut, B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations 66 (1987) 118-139]. In particular, we show that the exponential rate of decay is uniform in bounded sets of initial data.     

On recent developments in the theory of abstract differential equations with fractional derivatives

This note is motivated from some recent papers treating the problem of the existence of a solution for abstract differential equations with fractional derivatives. We show that the existence results in [Agarwal et al. (2009) [1], Belmekki and Benchohra (2010) [2], Darwish et al. (2009) [3], Hu et al. (2009) [4], Mophou and N’Guérékata (2009) [6] and [7], Mophou (2010) [8] and [9], Muslim (2009) [10], Pandey et al. (2009) [11], Rashid and El-Qaderi (2009) [12] and Tai and Wang (2009) [13]] are incorrect since the considered variation of constant formulas is not appropriate. In this note, we also consider a different approach to treat a general class of abstract fractional differential equations.     

A New Approach to Linearly Perturbed Riccati Equations Arising in Stochastic Control

In this paper a linearly perturbed version of the well-known matrix Riccati equations which arise in certain stochastic optimal control problems is studied. Via the concepts of mean square stabilizability and mean square detectability we improve previous results on both the convergence properties of the linearly perturbed Riccati differential equation and the solutions of the linearly perturbed algebraic Riccati equation. Furthermore, our approach unifies, in some way, the study for this class of Riccati equations with the one for classical theory, by eliminating a certain inconvenient assumption used in previous works (e.g., [10] and [26]). The results are derived under relatively weaker assumptions and include, inter alia, the following: (a) An extension of Theorem 4.1 of [26] to handle systems not necessarily observable. (b) The existence of a strong solution, subject only to the mean square stabilizability assumption. (c) Conditions for the existence and uniqueness of stabilizing solutions for systems not necessarily detectable. (d) Conditions for the existence and uniqueness of mean square stabilizing solutions instead of just stabilizing. (e) Relaxing the assumptions for convergence of the solution of the linearly perturbed Riccati differential equation and deriving new convergence results for systems not necessarily observable. Accepted 30 July 1996     

超声速、高超声速粘性气体分离流动的显隐式差分解法

本文利用反扩散的两步显、隐式差分方法,求解了超声速、高超声速粘性气体绕二维、三维压缩拐角的层流和湍流分离运动.结果表明,它既能获得很好的精度,又能大大缩短计算机时.     

Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations

The compactness of weak solutions to the magnetohydrodynamic equations for the viscous, compressible, heat conducting fluids is considered in both the three-dimensional space R³ and the three-dimensional periodic domains. The viscosities, the heat conductivity as well as the magnetic coefficient are allowed to depend on the density, and may vanish on the vacuum. This paper provides a different idea from [X. Hu, D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. (2008), in press] to show the compactness of solutions of viscous, compressible, heat conducting magnetohydrodynamic flows, derives a new entropy identity, and shows that the limit of a sequence of weak solutions is still a weak solution to the compressible magnetohydrodynamic equations.     

理想塑性问题中的一般方程、双调和方程和本征方程

本文推广了文[39]、[19]和[37]中关于理想塑性轴对称问题的结果,得到了三维理想塑性问题的一般方程。在引入量子电动力学中著名的Pauli矩阵后,本文以不同于文[7]的方法,使理想刚塑性材料的平面应变问题,最后归结为求解双调和方程。本文还以应力增量的偏张量为本征函数,导出了理想塑性问题的本征方程,从而使非线性成为线性方程的求解。     

Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps

A strong solutions approximation approach for mild solutions of stochastic functional differential equations with Markovian switching driven by Lévy martingales in Hilbert spaces is considered. The Razumikhin–Lyapunov type function methods and comparison principles are studied in pursuit of sufficient conditions for the moment exponential stability and almost sure exponential stability of equations in which we are interested. The results of [A.V. Svishchuk, Yu.I. Kazmerchuk, Stability of stochastic delay equations of Itô form with jumps and Markovian switchings, and their applications in finance, Theor. Probab. Math. Statist. 64 (2002) 167–178] are generalized and improved as a special case of our theory.     

Carleman estimates for one-dimensional degenerate heat equations

In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain. We derive new Carleman estimates for the degenerate parabolic equation $$ w_t + \left( {a\left( x \right)w_x } \right)_x = f,\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where the function a mainly satisfies $$ a \in \mathcal{C}^0 \left( {\left[ {0,1} \right]} \right) \cap \mathcal{C}^1 \left( {\left( {0,1} \right)} \right),a \gt 0 \hbox{on }\left( {0,1} \right) \hbox{and }\frac{1} {{\sqrt a }} \in L^1 \left( {0,1} \right). $$ We are mainly interested in the situation of a degenerate equation at the boundary i.e. in the case where a(0)=0 and / or a(1)=0. A typical example is a(x)=x^α (1 − x)^β with α, β ∈ [0, 2). As a consequence, we deduce null controllability results for the degenerate one dimensional heat equation $$ u_t - (a(x)u_x )_x = h\chi _w ,\quad (t,x) \in (0,T) \times (0,1),\quad \omega \subset \subset (0,1). $$ The present paper completes and improves previous works [7, 8] where this problem was solved in the case a(x)=x^α with α ∈[0, 2). Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

The time-varying discrete four block Nehari problem: A generalized Popov-Yakubovich type approach

The paper provides necessary and sufficient solvability conditions for the time-variant discrete four block Nehari problem in terms of the existence of the stabilizing solutions to two coupled Riccati equations. A parametrization of the class of all solutions is also given. The results are easily obtained from a signature condition — a generalized Popov Yakubovich type argument-imposed on an appropiate rational node. The present development may be seen as an alternative of the theory developed by Gohberg, Kaashoek and Woerdeman [15].     

Fire containment in grids of dimension three and higher

We consider a deterministic discrete-time model of fire spread introduced by Hartnell [Firefighter! an application of domination, Presentation, in: 20th Conference on Numerical Mathematics and Computing, University of Manitoba in Winnipeg, Canada, September 1995] and the problem of minimizing the number of burnt vertices when a fixed number of vertices can be defended by firefighters per time step. While only two firefighters per time step are needed in the two-dimensional lattice to contain any outbreak, we prove a conjecture of Wang and Moeller [Fire control on graphs, J. Combin. Math. Combin. Comput. 41 (2002) 19-34] that 2d-1 firefighters per time step are needed to contain a fire outbreak starting at a single vertex in the d-dimensional square lattice for d?3; we also prove that in the d-dimensional lattice, d?3, for each positive integer f there is some outbreak of fire such that f firefighters per time step are insufficient to contain the outbreak. We prove another conjecture of Wang and Moeller that the proportion of elements in the three-dimensional grid P_n×P_n×P_n which can be saved with one firefighter per time step when an outbreak starts at one vertex goes to 0 as n gets large. Finally, we use integer programming to prove results about the minimum number of time steps needed and minimum number of burnt vertices when containing a fire outbreak in the two-dimensional square lattice with two firefighters per time step.     

Symmetric duality for a class of nondifferentiable multi-objective fractional variational problems

We introduce a symmetric dual pair for a class of nondifferentiable multi-objective fractional variational problems. Weak, strong, converse and self duality relations are established under certain invexity assumptions. The paper includes extensions of previous symmetric duality results for multi-objective fractional variational problems obtained by Kim, Lee and Schaible [D.S. Kim, W.J. Lee, S. Schaible, Symmetric duality for invex multiobjective fractional variational problems, J. Math. Anal. Appl. 289 (2004) 505-521] and symmetric duality results for the static case obtained by Yang, Wang and Deng [X.M. Yang, S.Y. Wang, X.T. Deng, Symmetric duality for a class of multiobjective fractional programming problems, J. Math. Anal. Appl. 274 (2002) 279-295] to the dynamic case.     

Stability by the linear approximation for discrete equations

This paper is concerned with exponential stability of solutions of perturbed discrete equations. For a given m>1 we will provide necessary and sufficient conditions for exponential stability of all perturbed systems with perturbation of order m under the assumption that the unperturbed linear system is exponentially stable. Basing on this result we obtained necessary and sufficient conditions for exponential stability of the perturbed system for all perturbations of order m>1 for regular systems. Our results are expressed in terms of regular coefficients of the unperturbed system.     

A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator

Let A be a possibly unbounded skew-adjoint operator on the Hilbert space X with compact resolvent. Let C be a bounded operator from D(A) to another Hilbert space Y. We consider the system governed by the state equation with the output y(t)=Cz(t). We characterize the exact observability of this system only in terms of C and of the spectral elements of the operator A. The starting point in the proof of this result is a Hautus-type test, recently obtained in Burq and Zworski (J. Amer. Soc. 17 (2004) 443-471) and Miller (J. Funct. Anal. 218 (2) (2005) 425-444). We then apply this result to various systems governed by partial differential equations with observation on the boundary of the domain. The Schrödinger equation, the Bernoulli-Euler plate equation and the wave equation in a square are considered. For the plate and Schrödinger equations, the main novelty brought in by our results is that we prove the exact boundary observability for an arbitrarily small observed part of the boundary. This is done by combining our spectral observability test to a theorem of Beurling on nonharmonic Fourier series and to a new number theoretic result on shifted squares.     

Existence of partially regular weak solutions to Landau-Lifshitz-Maxwell equations

In this paper, the authors establish the existence of partially regular weak solutions to the Landau-Lifshitz equations coupling with static Maxwell systems in 3 dimensions by Ginzburg-Landau approximation. It is proved that the Hausdorff measure of the singular set is locally finite. This extends the similar results of Ding and Guo [S. Ding, B. Guo, Hausdorff measure of the singular set of Landau-Lifshitz equations with a nonlocal term, Comm. Math. Phys. 250 (1) (2004) 95-117] from the stationary solutions to weak solutions and the results of Wang [C. Wang, On Landau-Lifshitz equations in dimensions at most four, Indiana Univ. Math. J. 55 (5) (2006) 1615-1644] from Landau-Lifshitz equations to Landau-Lifshitz-Maxwell equations.     

On the applicability of the Adomian method to initial value problems with discontinuities

In this paper we extend our results of L. Casasús, W. Al-Hayani [The decomposition method for ordinary differential equations with discontinuities, Appl. Math. Comput. 131 (2002) 245–251] to initial value problems with several types of discontinuities, giving relevant examples of linear and nonlinear cases.     

Centroaffine Bernstein problems

C.P. Wang [Geom. Dedicata 51 (1994) 63–74] studied the Euler–Lagrange equation for the centroaffine area functional of hypersurfaces. In terms of a local representation of the hypersurface as a graph, this equation is a complicated, strongly non-linear fourth order PDE. We consider classes of solutions satisfying these equations together with completeness conditions. We also formulate appropriate centroaffine Bernstein problems and give partial solutions.