We have a stochastic dynamic model: Xk+1 =f(Xk,uk,wk ). We can design a cost function to be optimized using dynamic programming algorithm. How do we design a cost function for this dynamic system to ensure stability?

In Chapter 4 of Ref. [a] for a quadratic cost function and a linear system (Xk+1 =AXk+Buk+wk), a proposition shows that under a few assumptions, the quadratic cost function results in a stable fixed state feedback. However, I think about how we can consider stability issue in the designation of the cost function as a whole when we are going to define the optimal control problem for a nonlinear system generally. Can we use the meaning of stability to design the cost function? Please share me your ideas.

[a] Bertsekas, Dimitri P., et al. Dynamic programming and optimal control. Vol. 1. No. 2. Belmont, MA: Athena scientific, 1995.

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