By Marc Mangel

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6 The following equations are a model for forest growth (Clark, 1976): dX/dt = at-bXe-'X, X(t,) =X,. 46) Here X ( t ) is the volume of timber of an age greater than or equal to to and a, b, and c are constants. 47) where ~ ( tx), is a known function. Assume that measurements are taken according to z(tk) = h(tk)X(tk) + 'k. (15 4 8 ) If c = 0, then the linear filtering equations apply. Otherwise, the nonlinear algorithms must be used. 5. 7 Solve and compare the filter equations in the linear (c = 0) and linearized (that is, when Xe-" is Taylor-expanded) cases.

12) The form of the objective functional shows that the object of control in this example is to get the process near the origin and keep it there with a minimum use of the control. 6) becomes 0 =J, + min(M(t)x’ + N(t)u2 + J,(A(~)x + B(t)u)+ +(t)~,>. 13). Then the minimizing u satisfies u*(t) = - JXB(t)/2N(t). 13) and simplifying gives the nonlinear equation 0 =J, + M(t)x2 + J,(A(t)x) - B2(t)J:/4N(t) + &(t)J,,. 13) was linear, but involved an optimization step. 15) is nonlinear, but contains no optimization step.

If A4 = 0, set f ( t ) = l/K(t). 16), works because of the special linear-quadratic structure of the problem. In general, such special structure is missing and the solution is harder to obtain. 4: Portfolio Selection Merton (1971) studies the following problem of optimal portfolio selection. There are two assets with prices pl, p 2 that change according to dp, = ply dt, d p , = p2(a d t + 0 dW). 22) The first asset is riskless; its price grows exponentially with growth parameter r. The second asset is risky.