 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-4/30k/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

u^{*} (x, t) = - \frac{F _{x} ( x , t )}{2 c _{t}} .

F (x, t) = a (t) x^{2}

. The equation and terminal condition become

a^{'} (t) = \frac{a ( t ) ^{2}}{c _{t}}, a (h) = 1.

This Riccati equation is separable and gives

\frac{1}{a ( t )} = 1 + \int_{t}^{h} \frac{d s}{c _{s}} .

Consequently the general optimal feedback law is

u^{*} (x, t) = - \frac{x}{c _{t} ( 1 + \int _{t}^{h} d s / c _{s} )} .

When

c_{t} = c

is constant,

F (x, t) = \frac{c}{c + h - t} x^{2}, u^{*} (x, t) = - \frac{x}{c + h - t} .

Along the corresponding optimal trajectory,

x_{s} = x (c + h - s) / (c + h - t)

, so the open-loop control is constant:

u_{s}^{*} = - x / (c + h - t)

for

s \in [t, h]

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