1. How to solve adjoint system for given y R n?
To solve the adjoint system for a given y R n, follow these steps: 1. Solve the adjoint system (2.8) using the given y R n. 2. Define ps(t) as the solution, where ps(y, t) represents the solution for a specific y and time t. 3. Find the optimal control z(t) by minimizing the objective function min uU ps(t), B(t)u at each moment of t [t 0 , T]. 4. Solve system (1.2) for u(t) = z(y, t) to find the solution x(t). 5. Compute x(T) by substituting t = T into the solution x(t). 6. Calculate P(y) using the formula P(y) = ph(y), x(T) - y, where ph(y) represents the value of the function ph at y. This definition was introduced in reference [6]. Lemma 3 states that if there exists a feasible point z D and a point y i A m z such that ph(y j), u j - y j > 0, then ph(u j) > ph(z). The proof follows from Lemma 2, utilizing the quasiconvexity of ph(*) and global optimality conditions. The algorithm proposed for solving problem (1.7) differs from Algorithm 2 [6] in finding a local optimal control.
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2. What is the purpose of Step 1 in Algorithm OPTGL?
Step 1 initializes the control variable k to 0 and sets an arbitrary control uk V. It then finds a local optimal control u k using the optimal control software OPTCON. This step is crucial for starting the algorithm and determining the initial control value.
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