Maximum principle and second-order conditions for minimax problems of optimal control

In this paper, we study the optimal control problem of minimizing the functionalJ(x, u)=max_t1tt2(x(t),t). We formulate and prove necessary optimality conditions for this problem. We establish the equivalence between the initial minimax problem and a problem involving a terminal functional and phase constraints.     

A maximum principle for fourth order ordinary differential equations

We present a maximum principle for fourth order ordinary differential equations, based on a new approach involving counting of inflection points. We use our results to compute solutions of nonlinear equations describing static displacements of a uniform beam     

Maximum principle at infinity for complete minimal surfaces in flat 3-manifolds

The main result of this paper is the following maximum principle at infinity:Theorem.Let M ₁ and M ₂ be two disjoint properly embedded complete minimal surfaces with nonempty boundaries, that are stable in a complete flat 3-manifold. Then dist(M ₁,M ₂)=min(dist(M ₁,M ₂), dist(M ₂,M ₁)).In case one boundary is empty, e.g. M ₁,then dist(M ₁,M ₂)=dist(M ₂,M ₁).If both boundaries are empty, then M ₁ and M ₂ are flat.     

A similarity principle for complex vector fields and applications

This paper establishes a similarity principle for a class of non-elliptic, smooth complex vector fields in the plane. This principle is used to prove a uniqueness result for a nonlinear Cauchy problem.

     

The compact support principle for differential inequalities with gradient terms

We give some sufficient conditions to guarantee the validity of the Compact Support Principle for solutions of some singular elliptic differential inequalities on complete, non-compact manifolds. Our results apply in particular to the p-Laplace and the Mean Curvature operators. We also show that our results are sharp in many important cases.     

Maximum and antimaximum principles for a second order differential operator with variable coefficients of indefinite sign

New criteria for the existence of a maximum or antimaximum principle of a general second order operator with periodic conditions, as well as conditions for nonresonance, are provided and compared with the related literature.     

Maximum principle for optimal distributed control of the viscous Dullin-Gottwald-Holm equation

In this paper, the optimal distributed control of the viscous Dullin-Gottwald-Holm equation is investigated. Adopting the Dubovitskii and Milyutin functional analytical approach, we obtain the Pontryagin maximum principle of the system. The necessary optimality condition is established for an optimal control problem in fixed final horizon case. Finally, an illustrative example is also given.     

Maximum principle for the generalized time-fractional diffusion equation

In the paper, a maximum principle for the generalized time-fractional diffusion equation over an open bounded domain is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Caputo-Dzherbashyan fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial-boundary-value problem for the generalized time-fractional diffusion equation possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions.     

Maximum principle of optimal control for functional differential systems

We show that the maximum principle holds for optimal periodic control problems governed by functional differential equations under a Lipschitz condition on the value functional. Generalizations to other boundary conditions are also considered.This research was partially supported by NSF Grant No. DMS-84-01719.The first author was partially supported by the Science Fund of the Chinese Academy of Sciences, Beijing, China.     

Optimal Hardy-Rellich inequalities, maximum principle and related eigenvalue problem

In this paper we deal with three types of problems concerning the Hardy-Rellich's embedding for a bi-Laplacian operator. First we obtain the Hardy-Rellich inequalities in the critical dimension n=4. Then we derive a maximum principle for fourth order operators with singular terms. Then we study the existence, non-existence, simplicity and asymptotic behavior of the first eigenvalue of the Hardy-Rellich operator under various assumptions on the perturbation q.     

A new maximum principle of elliptic differential equations in divergence form

In this paper will be presented a new maximum principle of elliptic differential equations in divergence form which can be regarded as the counterpart of the Alexandroff-Bakelman-Pucci maximum principle of elliptic differential equations in nondivergence form.

     

The estimate for number of zeros of solutions of second order functional differential equations

In this paper, by extending the maximum principle, we study the number of zeros of solutions of second order functional differential equations. We obtain a sufficient condition for the existence of at most one zero of solutions on an interval. On this basis, we estimate the maximal number of zeros of solutions on a large interval. For illustrating the theoretical analysis, we also give two numerical simulation examples.     

Maximum principle for Hadamard fractional differential equations involving fractional Laplace operator

The purpose of the current study is to investigate IBVP for spatial-time fractional differential equation with Hadamard fractional derivative and fractional Laplace operator(−Δ)^β. A new Hadamard fractional extremum principle is established. Based on the new result, a Hadamard fractional maximum principle is also proposed. Furthermore, the maximum principle is applied to linear and nonlinear Hadamard fractional equations to obtain the uniqueness and continuous dependence of the solution of the IBVP at hand.     

On a Liouville-type comparison principle for solutions of semilinear elliptic partial differential inequalities

This Note is devoted to the study of a Liouville-type comparison principle for entire weak solutions of semilinear elliptic partial differential inequalities of the form $Lu+{|u|}^{q?1}u?Lv+{|v|}^{q?1}v$, where $q>0$ is a given number and L is a linear (possibly non-uniformly) elliptic partial differential operator of second order in divergent form given formally by the relation

$L=\sum _{i,j=1}^n\frac{?}{?x_i}[a_{ij}(x)\frac{?}{?x_j}].$

We assume that $n?2$, that the coefficients $a_{ij}(x)$, $i,j=1,\dots ,n$, are measurable bounded functions on ${\mathbb{R}}^n$ such that $a_{ij}(x)=a_{ji}(x)$, and that the corresponding quadratic form is non-negative. The results obtained in this work complete similar results on solutions of quasilinear elliptic partial differential inequalities announced in Kurta [C. R. Acad. Sci. Paris, Ser. I 336 (11) (2003) 897–900]. To cite this article: V.V. Kurta, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

A note on the strong maximum principle and the compact support principle

In this note we are concerned with the strong maximum principle (SMP) and the compact support principle (CSP) for non-negative solutions to quasilinear elliptic inequalities of the form

     

A remark on the maximum principle and stochastic completeness

We prove that the stochastic completeness of a Riemannian manifold is equivalent to the validity of a weak form of the Omori-Yau maximum principle. Some geometric applications of this result are also presented.

     

Maximum principles and the method of upper and lower solutions for time-periodic problems of the telegraph equations

This paper deals with the existence of bounded time-periodic solutions for the nonlinear telegraph equation

     

Maximum principle for the optimal control of a hyperbolic equation in one space dimension,part 2: Application

The optimal open-loop control of a beam subject to initial disturbances is studied by means of a maximum principle developed for hyperbolic partial differential equations in one space dimension. The cost functional representing the dynamic response of the beam is taken as quadratic in the displacement and its space and time derivatives. The objective of the control is to minimize a performance index consisting of the cost functional and a penalty term involving the control function. Application of the maximum principle leads to boundary-value problems for hyperbolic partial differential equations subject to initial and terminal conditions. The explicit solution of this system is obtained yielding the expressions for the state and optimal control functions. The behavior of the controlled and uncontrolled beam is studied numerically, and the effectiveness of the proposed control is illustrated.