Harald Kirchsteiger Thank you for the answer. In contrast to your viewpoint, consider the temperature in a room reached the value Temperature=0 (Celsius). In case the control action (heating/cooling power) does not stop, then the Temperature would not stay at that level for a long time, as nonzero control (heating/cooling) would change the stable temperature to higher/lower degrees. Hence, at any interval of temperature control (control operation), the heating/cooling should stop as the temperature reaches steady-state desired value. However, after a considerable time, control action (heating/cooling) should again be activated. But for the steady state solution (a specific interval of control), as the temperature reaches its desired value in steady state, control should be interrupted. Assume the ordinary differential equation xdot=x^3+u. When xdot approaches zero (steady-state situation) u should go to zero, and hence based on the ode, x should also go to zero. If u would not cease action, x and xdot would alter and oscillate.

Moreover, I am interested to know this idea with regard to optimal control. Does this assumption hold for optimal control problems (?) where velocity at the end of operation is not necessarily zero.

Hey Saeb AmirAhmadi Chomachar,

Here is a simple example of a double integrator with a generic PD controller, which shows that the control signal u crosses zero at least twice when the states are non-zeros.

x'1 = x2

x'2 = u

u = − (2/√2)*x2 − x1

with the initial conditions: x1(0) = 1, and x2(0) = 0.

By the way, you didn't mention about steady-state in your question.

Yew-Chung Chak Thank you. I missed to mention steady state phase in my question. By limit, I meant when x approaches zero with convergence notion and in a certain interval (not abruptly crossing the time axis). However, you are right with your objection, as your counterexample is clarifying. Moreover, I said my focus is optimal control, and as I know, in optimal control solution, state variables critically damp down to zero, and for non-zero initial conditions, state variable becomes zero only at steady-state (when t gets infinite), and therefore, u becomes zero at only such a steady-state zero state variable. Hence, I guess at optimal solution, limit (u(x)) is zero when x goes to zero, hence, probably we can conclude u(0)=0, as is lucid in plots (u versus x) for optimal regulation in existing benchmarks.

But my concern, is to know whether this assumption is valid, when optimal control is used for trajectory optimization (when final state and possibly its rate are not zero). However, I think for the case of trajectory optimization as lunar landing, we can still assume u(x=0)=u(0)=0, whereas this happens at the initiation of optimal control operation, whilst initial state is probably zero. But I am still looking to have more evidence for such optimal control problems and other free end-time (tf; free), and whether the assumption u(x=0)=0, holds or not? Notice that in optimal control, usually control variable is a function of time u=u(t), and not a feedback-law as u=u(x).