The aim of this paper is to give a detailed analysis of Hopf bifurcation of a ratio-dependent predator–prey system involving two discrete delays. A delay parameter is chosen as the bifurcation parameter for the analysis. Stability of the bifurcating periodic solutions is determined by using the centre manifold theorem and the normal form theory introduced by Hassard et al. Some of the bifurcation properties including the direction, stability and period are given. Finally, our theoretical results are supported by some numerical simulations.

We study the reflection of membrane-coupled gravity waves in deep water against a vertical barrier with a gap. A floating membrane is attached on both sides of the barrier. The associated mixed boundary value problem, which is not particularly well posed, is analysed. We utilize an orthogonal mode-coupling relation to reduce the problem to solving a set of dual integral equations with trigonometric kernel. We solve these by using a weakly singular integral equation. The reflection coefficient is determined explicitly, while having freedom to clamp the membrane with a spring of a certain stiffness on only one side of the vertical barrier. The physical problem is of capillary–gravity wave scattering by a vertical barrier with a gap, when the membrane density is neglected. In this case, the reflection coefficient is known up to an undetermined edge slope on either side of the barrier. The scattering quantity is computed and presented graphically against a wave parameter for different values of nondimensional parameters pertaining to the structures involved in the problem.

This paper is addressed to proving a new Carleman estimate for stochastic parabolicequations. Compared to the existing Carleman estimate in this respect (see [S. Tang and X.Zhang, SIAM J. Control Optim. 48 (2009) 2191–2216.], Thm.5.2), one extra gradient term involving in that estimate is eliminated. Also, our improvedCarleman estimate is established by virtue of the known Carleman estimate fordeterministic parabolic equations. As its application, we prove the existence ofinsensitizing controls for backward stochastic parabolic equations. As usual, thisinsensitizing control problem can be reduced to a partial controllability problem for asuitable cascade system governed by a backward and a forward stochastic parabolicequation. In order to solve the latter controllability problem, we need to use ourimproved Carleman estimate to establish a suitable observability inequality for somelinear cascade stochastic parabolic system, while the known Carleman estimate for forwardstochastic parabolic equations seems not enough to derive the desired inequality.

In this paper, we investigate the controllability of an underwater vehicle immersed in aninfinite volume of an inviscid fluid whose flow is assumed to be irrotational. Taking ascontrol input the flow of the fluid through a part of the boundary of the rigid body, weobtain a finite-dimensional system similar to Kirchhoff laws in which the control inputappears through both linear terms (with time derivative) and bilinear terms. ApplyingCoron’s return method, we establish some local controllability results for the positionand velocities of the underwater vehicle. Examples with six, four, or only three controlsinputs are given for a vehicle with an ellipsoidal shape.

The subject of this paper is the rigorous derivation of reduced models for a thin plate by means of Γ-convergence, in the framework of finite plasticity. Denoting by ε the thickness of the plate, we analyse the case where the scaling factor of the elasto-plastic energy per unit volume is of order ε2α−2, with α ≥ 3. According to the value of α, partially or fully linearized models are deduced, which correspond, in the absence of plastic deformation, to the Von Kármán plate theory and the linearized plate theory.

We consider the problem of minimizing \hbox{$\int_{0}^\ell \sqrt{\xi^2 +K^2(s)}\, {\rm d}s $}${{}^{\mathrm{\int }}}_{\mathrm{0}}^{\mathit{\ell }}\sqrt{{\mathit{\xi }}^{\mathrm{2}}\mathrm{+}{\mathit{K}}^{\mathrm{2}}\mathrm{\left(}\mathit{s}\mathrm{\right)}}\hspace{0.17em}\mathrm{d}\mathit{s}$ for a planar curve having fixedinitial and final positions and directions. The total length ℓ is free. Heres is thearclength parameter, K(s) is the curvature of the curveand ξ > 0 is a fixed constant. This problem comes from a model of geometry ofvision due to Petitot, Citti and Sarti. We study existence of local and global minimizersfor this problem. We prove that if for a certain choice of boundary conditions there is noglobal minimizer, then there is neither a local minimizer nor a geodesic. We finally giveproperties of the set of boundary conditions for which there exists a solution to theproblem.

In the paper we investigate the regularity of the value function representing Hamilton–Jacobi equation: − Ut + H(t, x, U, − Ux) = 0 with a final condition: U(T,x) = g(x). Hamilton–Jacobi equation, in which the Hamiltonian H depends on the value of solution U, is represented by the value function with more complicated structure than the value function in Bolza problem. This function is described with the use of some class of Mayer problems related to the optimal control theory and the calculus of variation. In the paper we prove that absolutely continuous functions that are solutions of Mayer problem satisfy the Lipschitz condition. Using this fact we show that the value function is a bilateral solution of Hamilton–Jacobi equation. Moreover, we prove that continuity or the local Lipschitz condition of the function of final cost g is inherited by the value function. Our results allow to state the theorem about existence and uniqueness of bilateral solutions in the class of functions that are bounded from below and satisfy the local Lipschitz condition. In proving the main results we use recently derived necessary optimality conditions of Loewen–Rockafellar [P.D. Loewen and R.T. Rockafellar, SIAM J. Control Optim. 32 (1994) 442–470; P.D. Loewen and R.T. Rockafellar, SIAM J. Control Optim. 35 (1997) 2050–2069].

A deterministic affine-quadratic optimal control problem is considered. Due to the nature of the problem, optimal controls exist under some very mild conditions. Further, it is shown that under some assumptions, the optimal control is unique which leads to the differentiability of the value function. Therefore, the value function satisfies the corresponding Hamilton–Jacobi–Bellman equation in the classical sense, and the optimal control admits a state feedback representation. Under some additional conditions, it is shown that the value function is actually twice differentiable and the so-called quasi-Riccati equation is derived, whose solution can be used to construct the state feedback representation for the optimal control.