Optimal control problems in spaces of functions of bounded variation. (English) Zbl 0879.49003

The author investigates the minimization problem for a Bolza-type cost functional \[ \ell(x(a),x(b))+\int_a^bf(t,x,v)dt \] under the constraint formed by the state equation \(dx/dt=g(t,x)+h(t,x)v\) where \(f(t,x,\cdot)\) is convex but has a linear growth only. Supposing coercivity, minimizing sequences \((x_k,v_k)\) are thus bounded only in the non-reflexive space \(W^{1,1}(a,b;\mathbb R^N)\times L^1(a,b;\mathbb R^m)\). Thus the problem need not have any solution due to concentration effects. A relaxation by an extension on the BV-space is performed. As \(h(t,\cdot)\) is nonconstant, a complicated interplay between the possible jumps of \(x(\cdot)\) and the concentrations (impulses) in \(u\) can occur and thus the cost functional cannot be extended independently of the state equation. This makes the problem and the used techniques very nontrivial. A comparison with already known results is provided, too.


49J15 Existence theories for optimal control problems involving ordinary differential equations
49N25 Impulsive optimal control problems