Perimeter Estimates for Attainable Sets in Control Theory

The reachable set at time T>0 from a given closed set K⊂Rⁿ, A(T;K), is a well known object in control theory. Here such a set is investigated for the symmetric system ẋ(t)=f(x(t))u(t), u(t)∈B̄ A recent result obtained by the first author in collaboration with Frankowska guarantees that, under suitable assumptions, A(K;T) satisfies a uniform interior sphere condition for T>0. Using such a property we show that, for f(x) smooth and non degenerate, A(K;T) has finite perimeter, and we obtain sharp estimates for the time-dependence of the perimeter and volume of such a set.