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Singularities of the minimum time function for semilinear parabolic systems. (English) Zbl 0916.49020

The system is \[ \begin{aligned}{\partial y(t, x) \over \partial t} &= \Delta y(t, x) + f(y(t, x)) + u(t, x) \quad (t > 0, \;x \in \Omega),\\ y(0, x) &= \zeta(x) \quad (x \in \Omega),\end{aligned} \] with suitable boundary conditions, in an \(n\)-dimensional domain \(\Omega\) with smooth boundary \(\Gamma\). The solutions \(y(t, \zeta, u, \cdot)\) and the control strategy \(u(t,\cdot)\) take values in \(H = L^2(0, T;\mathbb{R}^m)\) and the control strategy obeys the constraint \[ \int_\Omega | u(t, x)| ^2 dx \leq R^2 \quad (t \geq 0). \] The function \(f\) satisfies \(f(0) = 0\) and is continuously differentiable with \(\nabla f\) bounded and Lipschitz continuous.
The authors discuss the dynamic programming approach to the problem of driving the initial condition \(\zeta(\cdot)\) to the ball \(B_r \subset H\) of center \(0\) and radius \(r\) in minimum time. The value function for this problem is the minimum time function \[ T(\zeta) = \inf_u \inf_t \{t \geq 0; \| y(t, \zeta, u, \cdot)\| \leq r \}. \] This function solves the Hamilton-Jacobi equation \[ R\| DT(\zeta)\| - \langle A\zeta + F(\zeta), DT(\zeta) \rangle = 1 \] where \(\zeta\) belongs to the controllable set \({\mathcal R}\) of all \(\zeta \in H\) that can be driven to \(B_r\) in finite time and \(F : H \to H\) is the nonlinear operator \(F(y)(x) = f(y(x))\). At the present time, the Hamilton-Jacobi equation is reasonably well understood in the linear case \(F = 0\) (see references in this paper), but it is much more difficult to handle when the equation is nonlinear due to the fact that \(T\) may develop singularities; these singularities are related to nonuniqueness of optimal trajectories, a typical nonlinear phenomenon. After establishing the connection between singularities and nonuniqueness, the authors provide an example of a \(2 \times 2\) reaction-diffusion system with multiple time optimal trajectories, thus with a singular minimum time function.

MSC:

49L20 Dynamic programming in optimal control and differential games
35K55 Nonlinear parabolic equations
49J20 Existence theories for optimal control problems involving partial differential equations
93C20 Control/observation systems governed by partial differential equations
49L99 Hamilton-Jacobi theories
35K57 Reaction-diffusion equations
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