Singular Optimal Control of a 1-D Parabolic-Hyperbolic Degenerate Equation
نویسندگان
چکیده
In this paper, we consider the controllability of a strongly degenerate parabolic equation with a degenerate one-order transport term. Despite the strong degeneracy, we prove a result of well-posedness and null controllability with a Dirichlet boundary control that acts on the degenerate part of the boundary. Then, we study the uniform controllability in the vanishing viscosity limit and prove that the cost of the control explodes exponentially fast in small time and converges exponentially fast in large time in some adapted weighted norm. The main tools used are a spectral decomposition involving Bessel functions and their zeros, some usual results on admissibility of scalar controls for diagonal semigroups, and the moment method of Fattorini and Russell.
منابع مشابه
A note on critical point and blow-up rates for singular and degenerate parabolic equations
In this paper, we consider singular and degenerate parabolic equations$$u_t =(x^alpha u_x)_x +u^m (x_0,t)v^{n} (x_0,t),quadv_t =(x^beta v_x)_x +u^q (x_0,t)v^{p} (x_0,t),$$ in $(0,a)times (0,T)$, subject to nullDirichlet boundary conditions, where $m,n, p,qge 0$, $alpha, betain [0,2)$ and $x_0in (0,a)$. The optimal classification of non-simultaneous and simultaneous blow-up solutions is determin...
متن کاملMildly degenerate Kirchhoff equations with weak dissipation: global existence and time decay
We consider the hyperbolic-parabolic singular perturbation problem for a degenerate quasilinear Kirchhoff equation with weak dissipation. This means that the coefficient of the dissipative term tends to zero when t → +∞. We prove that the hyperbolic problem has a unique global solution for suitable values of the parameters. We also prove that the solution decays to zero, as t → +∞, with the sam...
متن کاملThe geometric properties of a degenerate parabolic equation with periodic source term
In this paper, we discuss the geometric properties of solution and lower bound estimate of ∆um−1 of the Cauchy problem for a degenerate parabolic equation with periodic source term ut =∆um+ upsint. Our objective is to show that: (1)with continuous variation of time t, the surface ϕ = [u(x,t)]mδq is a complete Riemannian manifold floating in space RN+1and is tangent to the space RN at ∂H0(t); (2...
متن کاملOptimal Control for Degenerate Parabolic Equations
This paper considers the optimal control of a degenerate parabolic partial differential equation governing a di usive population with logistic growth terms. Assuming this population causes damage to forest and agricultural land, the optimal control is the trapping rate and the cost functional is a combination of the damage and trapping costs. We prove existence, uniqueness, and regularity resul...
متن کاملHyperbolic–parabolic singular perturbation for nondegenerate Kirchhoff equations with critical weak dissipation
We consider the hyperbolic-parabolic singular perturbation problem for a nondegenerate quasilinear equation of Kirchhoff type with weak dissipation. This means that the dissipative term is multiplied by a coefficient b(t) which tends to 0 as t→ +∞. The case where b(t) ∼ (1 + t) with p < 1 has recently been considered. The result is that the hyperbolic problem has a unique global solution, and t...
متن کامل