Continuous dependence with respect to the input of trajectories of control-affine systems

We study the continuous dependence on the input of trajectories of control- affine systems belonging to the class C-0 (m) of all systems Sigma of the f orm [GRAPHICS] where f(o);...; f(m) are continuous vector fields on some open subset of R- n and the control functions belong to L-1 ([0; T]; R-m). We give a simple n ecessary and sufficient condition for a control sequence {u(j)}(j=1)(infini ty) to "T-o -converge" to a control u(infinity), i.e., to be such that, for every system Sigma in C-o(m), the trajectories generated by the u(j) conve rge as j --> infinity to the trajectories generated by u(infinity). We also characterize T-k-convergence (the concept of control convergence that aris es when we use, instead of C-o(m), the class C-k(m) of systems Sigma where the f(i) are of class C-k) for k greater than or equal to 1 in the scalar i nput case, and we explain how the analogous characterization for the multi- input case fails to be true, unless one restricts oneself to the class C-co mm(k) (m) of systems for which the vector fields f(1);...; f(m) commute. As a preliminary, we define a "topology of trajectory convergence" (or "T-con vergence") on the set of all time-varying vector fields Omega x I There Exi sts (x; t) bar right arrow f(x; t) is an element of R-n, where Omega is an open subset of R-n and I is an interval, and we study some of its propertie s. This enables us to make the definition of T-k-convergence precise for se quences and, more generally, for nets, by saying that a net {u(alpha)}(alph a is an element of A) in L-1 ([0; T]; R-m) T-k-converges to a limit u(infin ity) if for every system Sigma in C-k (m) the time-varying vector fields (x ; t) bar right arrow f(o)(x) + Sigma(i=1)(m) u(i)(alpha)(t)f(i)(x) T-k-conv erge to (x; t) bar right arrow f(o)(x) + Sigma(i=1)(m) u(i)(infinity)(t)f(i )(x).