Optimal control of the viscous Degasperis-Procesi equation

Abstract

This paper studies the problem for optimal control of the viscous Degasperis-Procesi equation. The existence and uniqueness of weak solution to the equation are proved in a short interval. The optimal control of the viscous Degasperis-Procesi equation under boundary condition is given and the existence of optimal solution to the equation is proved.

The optimal control problem with necessity condition for a viscous shallow water equation

“…Due to the independence of coefficients m, a and b in (1.4), the nonlinear term uu x does not disappear after using the transformation y = uu xx , which leads to the difficulty of establishing the estimates for term uu x . This is the major improvement in comparison with the results in the literature [5,10,17,32], where the problems studied are special cases of the optimal control problem (1.3) in this paper. Moreover, we obtain the necessity condition and local uniqueness of optimal control to the optimal control problem (1.3) by using the Gâteaux derivative of cost functional.…”

“…We replace u n , y n by u n k , y n k in (3.12), respectively. Taking k → ∞ shows that the limit function y satisfies 17) in the weak solution sense. From Theorem 2.1, we obtain the uniqueness of weak solutions to problem (3. .…”

“…The local well-posedness for the Cauchy problem of the DegasperisProcesi equation and blow-up mechanism of solutions were studied in [15,16]. Tian [17] investigated the optimal control problem for a viscous Degasperis-Procesi equation by using the Galerkin method and optimal control theory of distributed parameter system. However, the nonlinear partial differential equations created to model physical processes play an important role in almost all branches of mathematics.…”

On the Cauchy problem for the generalized shallow water wave equation

“…About the application of harmonic analysis in partial difference equation see [8][9][10][11][12][14][15][16]. About shallow water wave see [18][19][20][21][22][23][24][25][26][27][28].…”

Optimal Control of the Viscous Weakly Dispersive Benjamin-Bona-Mahony Equation

“…Smaoui [12] studied the boundary and distributed control of the viscous Burgers equation. L. Tian and C. Shen [15] considered the optimal control of the viscous Degasperis-Procesi equation.…”