obv : Optimization for a BVP.

This demo illustrates use of the method of successive continuation for a boundary value optimization problem. A detailed description of the basic method, as well as a discussion of the specific application considered here, is given in DoKeKe:91b DoKeKe:91b. The required extended system is fully programmed here in the user-supplied routines in obv.f. For the case of periodic solutions the optimality system can be generated automatically; see the demo ops.

Consider the system

$$\begin{array}{cl} u_1'(t) & = u_2(t), \\ u_2'(t) &= -\lambda_1 e^{p(u_1,\lambda_2,\lambda_3)}, \end{array}$$ (17.3)

where $p(u_1,\lambda_2,\lambda_3) \equiv u_1 + \lambda_2 u_1^{2} + \lambda_3 u_1^{4},$ with boundary conditions

$$\begin{array}{cl} u_1(0) &= 0, \\ u_1(1) &= 0. \\ \end{array}$$ (17.4)

The objective functional is

$$\omega = \int_0^{1} (u_1(t)-1)^{2}~ dt + \frac{1}{10} \sum_{k=1}^{3} \lambda_{k}^{2}.$$

The successive continuation equations are given by

$$\begin{array}{cl} u_1'(t) &= u_2(t), \\ u_2'(t) &= -\lambda_1... ...(t) + 2 \gamma(u_1(t)-1), \\ w_2'(t) &= -w_1(t), \\ \end{array}$$ (17.5)

where

$$p_{u_1} \equiv \frac{{\partial p} }{ {\partial u_1}} = 1 + 2\lambda_2 u_1 + 4\lambda_3 u_1^{3},$$

with

$$\begin{array}{cl} u_1(0) = 0,\qquad &w_1(0) - \beta_1 = 0,\qq... ...,\qquad &w_1(1) + \beta_2 = 0,\qquad w_2(1) = 0, \\ \end{array}$$ (17.6)

$$\int_0^{1} \bigl[ \omega - (u_1(t)-1)^{2} - \frac{1}{10} \sum_{k=1}^{3} \lambda_{k}^{2} \bigr]~ dt = 0,$$

$$\int_0^{1} \bigl[w_1^{2}(t) - \alpha_0 \bigr]~ dt = 0,$$

$$\begin{array}{cl} &\int_0^{1} \bigl[ -e^{p(u_1,\lambda_2,\lam... ...c{1}{5}\gamma \lambda_3 - \tau_3 \bigr]~ dt = 0. \\ \end{array}$$ (17.7)

In the first run the free equation parameter is $\lambda_1$. All adjoint variables are zero. Three extrema of the objective function are located. These correspond to branch points and, in the second run, branch switching is done at one of these. Along the bifurcating family the adjoint variables become nonzero, while state variables and $\lambda_1$ remain constant. Any such non-trivial solution point can be used for continuation in two equation parameters, after fixing the $L_2$-norm of one of the adjoint variables. This is done in the third run. Along the resulting family several two-parameter extrema are located by monotoring certain inner products. One of these is further continued in three equation parameters in the final run, where a three-parameter extremum is located.

Table 17.6: Commands for running demo obv.

AUTO -COMMAND	ACTION
! mkdir obv	create an empty work directory
cd obv	change directory
demo('obv')	copy the demo files to the work directory
run(c='obv.1')	locate 1-parameter extrema as branch points
sv('obv')	save output-files as b.obv, s.obv, d.obv
run(c='obv.2',s='obv')	compute a few step on the first bifurcating family. Constants changed : IRS, ISW, NMX
sv('1')	save the output-files as b.1, s.1, d.1
run(c='obv.3',s='1')	locate 2-parameter extremum; restart from s.1. Constants changed : IRS, ISW, NMX, ICP(3)
sv('2')	save the output-files as b.2, s.2, d.2
run(c='obv.4',s='2')	locate 3-parameter extremum; restart from s.2. Constants changed : IRS, ICP(4)
sv('3')	save the output-files as b.3, s.3, d.3