We study some nonlinear optimal control problems under state constraint. We construct extremal flows by differential-algebraic equations to solve an optimal control problem subject to mixed control-state constraint. Then we present an approximation approach to the state constraint optimal control problem.

Partial Differential Equation  Mathematics  Ordinary Differential Equation 

This paper is concerned with the normalized ground states to the following lower critical fractional Kirchhoff-Choquard type equations. Using the constraint variational method, we establish the existence of normalized ground states and analyze their asymptotic properties as as $\mu \to 0^{}$ or $c\to$...

Partial Differential Equation 

This paper is devoted to the normalized solutions of a planer L²-critical Schrödinger-Poisson system with an external potential V(x) =❘X❘²   and in-homogeneous attractive interactions K(x)∈(0,1). Applying the constraint variational method, we prove that the normalized solutions exist if and only if the interaction strength a satisfies a∈(0,a^*):=∥Q∥²_L²...

Partial Differential Equation  Mathematics 

Many kinds of explicit and exact solutions of the nonlinear Newell equation, including the solitary wave solution, the singular traveling wave solution, and the triangle function-type periodic wave solutions, are presented by a direct trial function approach.

Mathematical Analysis  Partial Differential Equation  Mathematics  Ordinary Differential Equation 

In this paper, we consider the following nonlinear Choquard equation -ε²Δw V(x)w=ε^{-θ(Y₁(w) Y₂(w))}, where ε>0, N>2, Y₁(w):=W₁(x)[I_θ*(W₁|w|^p)]|w|^p-2w, Y₂(w):=W₂(x)[I_θ*(W₂|w|^q)]|w|^q-2w, I_θ is the Riesz pote...

Partial Differential Equation 

In this paper, we study a nonlinear Schrödinger equation with competing potentials -ε²Δν V(x)ν=W₁(x)|ν|^p-2ν W₂(x)|ν|^q-2ν, ν∈H¹(R^N), where ε>0, p,q∈(2,2*), p>q, , V(x), W₁(x) and W₂(x) are continuous bounded positive functions. Under suitable assumptions on the potentials, we consider the existence, concentration, convergence and decay estima...

Partial Differential Equation 

In this study, we have determined Green’s functions for Helmholtz integral equations in a spherical polar coordinate system in the whole plane domain with the aid of spectral Fourier transform technique. Our intended Green’s function solution has a dominant role to represent wave propagation with a high quantum wave number. The Diracdelta function also plays an important role here to represent the scattering region for wave propagation. The evaluation of the improper double integrals in the comp...

Mathematical Analysis  Partial Differential Equation  Mathematics  Ordinary Differential Equation 

In this paper, we study the fractional Klein-Gordon-Maxwell system with steep potential well. On the basis of overcoming the lack of compactness, the ground state solution is obtained by proving that the solution satisfies the mountain pass level.

Partial Differential Equation 

In this paper, we study the nonautonomous Klein-Gordon-Maxwell system with logarithmic nonlinearity. We obtain the existence of nontrivial solution for this system by logarithmic Sobolev inequality and variational method.

Partial Differential Equation 

In this paper, we prove the existence of global strong solutions for the three-dimensional nonautonomous Brinkman-Forchheimer-extended-Darcy equation with singularly oscillating and show that the strong solutions are unique. In addition, we also give general estimates for its auxiliary linear equation; finally, we derive the oscillatory averaged estimates of the equation from the results of these general estimates.

Partial Differential Equation