The reachability problem for an equation of Lotka-Volterra type
The author considers the nonlinear control system \[ \dot x_i=x_i\left(x_i+\sum\limits_{j=1}^na_{ij}\ln | x_i| \right), \quad i=1,\ldots, n, \] \[ x=(x_1,\ldots x_n)\in {\mathbb R}^n,\, u\in P,\quad t_0\leq\tau\leq t,\, x (t_0)\in X_0, \] with \(X_0=\{x\in {\mathbb R}^n\mid (\ln | x| , Q\ln| x| )\leq 1\}\) and the objective function \[ V(t,x)=\min_{u\in P}\text{dist}(\ln| x_0| , X_0), \] where \(P=\{u\in {\mathbb R}^n\mid (u,P(t)u)\leq 1\}\). Using dynamic programming, the author studies the reachability set \(X(\tau)=\{x\in {\mathbb R}^n \mid V(\tau,x)\leq 0\} \) and constructs outer and inner guaranteed estimates for this set.
scientific article; zbMATH DE number 2099436