On a problem of the classification of a set of controls for a parabolic equation (Q1589162)

Let be given the controlled object: \[ {\partial v \over \partial t} (t,x)= {\partial^{2} v \over \partial x^{2}}(t,x)+u(t,x),\;t\in \theta=[0, T_{0}], x \in \Omega= (0,l), \] \[ v(0,x)=v_{0}(x)=0, v_{0} \in L_{2}(\Omega), \] \[ v(t,0)= v(t,l), t \in \theta. \] The admissible control function belongs to \[ {\mathcal U}=\left\{ u(\cdot , \cdot) \in L_{2}(\theta \times \Omega): u(t,\cdot) \in U, t\in \theta \right\} \] where \(U=\left\{ f\in L_{2}(\Omega):|f(x)|\leq 1, x \in \Omega\right\}\). For the investigated problem, the set of admissible controls may be defined in another way, i.e., \[ {\mathcal U}_{x}= \left\{ u(\cdot, \cdot) \in L_{2}(\theta \times \Omega): u(t, \cdot) \in U_{*}, t \in \theta \right\} \] where \(U_{*}=\left\{f \in L_{2}(\Omega):||f(\cdot)||_{L_{2}(\Omega)} \leq 1\right\}\). The problem is to classify the admissible control function. As a criterion of classification, there is understood the sign of the derivative of the minimum time function existing in the direction of the local shift of the controlled object.