Q2. What is the first optimal control problem?
Given the system (1) and the initial condition (2), find a combined element (u, τ ) ∈ G such that the cost functional (4), with the left hand side replaced by g0(u, τ ), is minimized over G.To solve Problem (P1) or (P2) numerically, the authors apply the classical control parametrization scheme.
Q3. What is the solution to the problem (P1)?
Given the system (10) and the initial condition (11), find a combined element (σ, θ) ∈ Υsuch that the cost function G0(σ, θ) is minimized over Υ subject toκi ∑l=1θl p = τi, i = 1, . . . ,m+ 1.Problem (P2(p)).
Q4. What is the simplest way to solve the drug delivery problem?
The authors choose n+1 fixed time points {ti} n i=0, where t0 = 0 and tn = T , and approximate the drug delivery rate by a constant σi on the interval [ti−1, ti), i = 1, . . . , n.
Q5. What is the funding for this research?
Research supported by the National Natural Science Foundation of China under Grant 60704003 and a grant from the Australian Research Council.
Q6. how can i recover characteristic times from (8)?
Then the characteristic times can be recovered from (8) byτi = t(ζi) =κi ∑l=1θl p , i = 1, . . . ,m+ 1. (9)Finally, applying the transformation to the dynamics (1) yieldsẏ(s) = vp(s)̃f (y(s),σ), (10)with the initial conditiony(0) = x0, (11)wherey(s) = x(t(s)) and f̃ (y(s),σ) = f (x(t(s)), ũp(s)).
Q7. What is the way to solve the optimal control problem?
Control parametrization and a time scaling transformation were applied to approximate this type of optimal control problem by a sequence of optimal parameter selection problems.
Q8. What is the way to choose parameters for the dynamic model?
The problem of choosing values for these unknown parameters in such a way that the solution of the dynamic model will best fit experimental data at a set of sample points can be formulated as an optimal parameter selection problem with the objective function depending on many characteristic times.
Q9. What is the advantage of the new scheme?
it has two important advantages over the existing scheme: the complexities involved in dealing with a discontinuous costate system are avoided; and, more importantly, no interpolation of the state is necessary if a variable step size integration method is used to solve the relevant differential equations.
Q10. What is the recent application of this technique?
This technique has recently been applied to the study of crystallization processes; see Livk, Pohar & Ilievski (1999) and Li, Livk & Ilievski (2001), for example.
Q11. How did they solve the fixed multiple characteristic time optimal control problem?
Martin & Teo (1994) and Martin (1992) solved the fixed multiple characteristic time optimal control problem numerically using the classical control parametrization scheme (see Teo, Goh & Wong (1991)).
Q12. What is the way to calculate the cost and constraint gradients?
Each of these optimal parameter selection problems can be viewed as a non-linear optimization problem and a new scheme for calculating the cost and constraint gradients was proposed.
Q13. What are the constraints on the amount of drug that can be administered?
These restrictions are expressed mathematically by the following two constraints on the drug concentration:0 ≤ v(t) ≤ vmax, for all t ∈ [0, T ], (33)and ∫ T0v(s)ds ≤ vcum, (34)where vmax and vcum are constants.