Lecture #7 Stability and convergence of ODEs Hybrid Control and

... The equilibrium point xeq ∈ Rn is (Lyapunov) stable if ∃ α ∈ 2: ||x(t) – xeq|| · α(||x(t0) – xeq||) ∀ t≥ t0≥ 0, ||x(t0) – xeq||· c Suppose we could show that ||x(t) – xeq|| always decreases along solutions to the ODE. Then ||x(t) – xeq|| · ||x(t0) – xeq|| ∀ t≥ t0≥ 0 we could pick α(s) = s ⇒ Lyapunov ...

... The equilibrium point xeq ∈ Rn is (Lyapunov) stable if ∃ α ∈ 2: ||x(t) – xeq|| · α(||x(t0) – xeq||) ∀ t≥ t0≥ 0, ||x(t0) – xeq||· c Suppose we could show that ||x(t) – xeq|| always decreases along solutions to the ODE. Then ||x(t) – xeq|| · ||x(t0) – xeq|| ∀ t≥ t0≥ 0 we could pick α(s) = s ⇒ Lyapunov ...

x - UCSB ECE

... The equilibrium point xeq 2 Rn is (Lyapunov) stable if 9 a 2 K: ||x(t) – xeq|| · a(||x(t0) – xeq||) 8 t¸ t0¸ 0, ||x(t0) – xeq||· c Theorem (Lyapunov): Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: Rn ! R such that Then xeq is a Lyapunov stable ...

... The equilibrium point xeq 2 Rn is (Lyapunov) stable if 9 a 2 K: ||x(t) – xeq|| · a(||x(t0) – xeq||) 8 t¸ t0¸ 0, ||x(t0) – xeq||· c Theorem (Lyapunov): Suppose there exists a continuously differentiable, positive definite, radially unbounded function V: Rn ! R such that Then xeq is a Lyapunov stable ...

On the optimal stopping of a one-dimensional diffusion Damien Lamberton Mihail Zervos

... The representation of differences of two convex functions given by (1.7) is also new. Such a result is important for the solution to one-dimensional infinite time horizon stochastic control as well as optimal stopping problems using dynamic programming. Indeed, the analysis of several explicitly sol ...

... The representation of differences of two convex functions given by (1.7) is also new. Such a result is important for the solution to one-dimensional infinite time horizon stochastic control as well as optimal stopping problems using dynamic programming. Indeed, the analysis of several explicitly sol ...

NBER WORKING PAPER SERIES OPTIMAL ENDOWMENT DESTRUCTION UNDER CAMPBELL-COCHRANE HABIT FORMATION Lars Ljungqvist

... endowment destruction must not drop below some treshold ψ, which generally depends on the current strategy ω and the current state. Next, assume that γ < 1. In that case, the loss from driving ψ̃0 → −∞ is finite and given by (exp ((1 − γ)(sa0 + ψ0 + y0 )) − 1)/(1 − γ). However, the first term in the ...

... endowment destruction must not drop below some treshold ψ, which generally depends on the current strategy ω and the current state. Next, assume that γ < 1. In that case, the loss from driving ψ̃0 → −∞ is finite and given by (exp ((1 − γ)(sa0 + ψ0 + y0 )) − 1)/(1 − γ). However, the first term in the ...