CAD.3:4.5 - Use a continuous-time branch only with its stronger premises

Let D be the domain of the connected continuous state on which the model and the bounds below apply, including the reference and any controller state on which the argument depends. Start the trajectory in D and suppose a differentiable function V satisfies, throughout D under the admitted input law,

V\geq c_e\|x-Pz\|^2,\qquad \dot V\leq-\alpha V+\beta\|w\|^2,

with positive c_e,\alpha, nonnegative \beta, and \|w(t)\|\leq\bar w. Let \tau_D be the first exit time from D, with \tau_D=\infty if there is no exit. For every 0\leq t<\tau_D on which the solution exists, integration gives

V(t)\leq e^{-\alpha t}V(0) +\frac{\beta\bar w^2}{\alpha}(1-e^{-\alpha t}).

Consequently \|x-Pz\|\leq\varepsilon on that interval, where \varepsilon^2=\max\{V(0),\beta\bar w^2/\alpha\}/c_e. If h is Lipschitz with constant L_h on a state domain containing both x(t) and Pz(t), requiring h(Pz(t))\geq L_h\varepsilon implies h(x(t))\geq0 on the same interval.

To claim the conclusion for all future time, separately establish continued existence and containment of the connected trajectory in D under the allowed reference evolution and disturbances, for example through a forward-invariant subset of D containing the admitted initial states. A tracking-error bound alone does not prevent the reference and plant from leaving the region where the derivative or Lipschitz bound is valid.

This is a sufficient construction for a particular connected dynamics and admissible input law. Obtain V and verify its derivative, domain, actuator feasibility and treatment of the moving reference; their existence is not supplied by the formula. Sampling, estimation, saturation and delay must either occur in that dynamics or be covered by its perturbation bound.