This file describes how to use a MACSYMA variational optimization
package to analytically solve problems from the calculus of variations
and the maximum principle, including optimal control.

Programmed by David R. Stoutemyer, (login name:STOUTE), Electrical
Engineering Department, University of Hawaii, August 1974.
Suggestions, questions, and interesting successful
or unsuccessful examples are welcome.

                        OTHER RELEVANT FILES:
OPTVAR >  is a MACSYMA batch file containing the functions and
   option settings for variational optimization.
OPTVAR LISP  is a LISP translation of OPTVAR >.
OPTVAR DEMO  is a MACSYMA batch file demonstrating various ways of
   using the optimization functions.
OPTVAR OUTPUT  is a text file containing  OPTVAR DEMO together with
   the output that it produces when executed.

                           USAGE:
In MACSYMA,  first type
             BATCH(OPTVAR, >, DSK, SHARE2);
or
             LOADFILE(OPTVAR, LISP, DSK, SHARE2);


Next, if interested in executing the demonstration, type

             LOADFILE(ODE2, FASL, DSK, SHARE);
             LOADFILE(DESOLN, LISP, DSK, SHARE);
then either
             BATCH(OPTVAR, DEMO, DSK, SHARE2);
or
             DEMO(OPTVAR, DEMO, DSK, SHARE2);

The latter alternative permits opportunities for interaction.

To derive the Euler-Lagrange equations for a calculus-of-variations
problem, type

             EL(F, YLIST, TLIST);

F  is an expression or the label of an expression for the integrand
   of the stationary functional, augmented by Lagrange multipliers
   times the integrands of any isoperimetric constraints and/or
   differential expressions constrained to equal zero.  The multipliers
   should be written as functions of the independent variables in the
   latter case.
YLIST  is a list of the dependent variables, or the label thereof.
TLIST  is a list of the independent variables, or the label thereof.

For convenience, square brackets may be omitted from 1-element lists.

EL displays one or more E-labeled equations, then returns a list
of the E-labels.  These equations are the Euler-Lagrange equations,
perhaps together with first integrals corresponding to conservation
of energy and/or conservation of momentum.  The former will contain a
constant of integration K[0], whereas the latter will contain constants
of integration K[I], with positive I.  The latter will immediately
follow the corresponding Euler-Lagrange equation.

OPTVAR DEMO or OPTVAR OUTPUT illustrates some ways that the
resulting differential equations may be solved analytically.


To derive the Hamiltonian and auxiliary differential equations for
an optimal control problem, type

             HAM(ODES);

ODES  is a list of the first-order differential equations that
   govern the state variables.  Each differential equation must be
   of the form
             'D(Y,T) = EXPRESSION
where  Y  is one of the dependent variables,  T is the independent
variable, and EXPRESSION  depends upon the independent, dependent, and
control variables.

HAM displays two or more E-labeled expressions, then returns a list
of the E-labels.  The first expression is the Hamiltonian, and the
other expressiona are the auxiliary diferential equations,
together with their general solutions,  AUX[I] = K[I], whenever the
Ith differential equation is of the trivial form  'D(AUX[I],T) = 0.
The K[I] are undetermined constants of integration.

HAM is directly suitable for the autonomous time-optimal problem.
Other problems may be converted to this form by introducing extra state
variables, as described in most optimal-control texts or in the report
referenced in  OPTVAR OUTPUT  and OPTVAR DEMO.

In simple enough cases, the routines in the SHARE files ODE2 FASL
and/or DESOLN LISP  may be used to determine an analytical
closed-form or series solution to the differential equations.  The
latter requires the dependencies to be explicitly exhibited, for
example:  'D(Y(T),T),; so  OPTVAR >  and  OPTVAR LISP  include a
function to perform this conversion.  To use it type*

             CONVERT(ODES, YLIST, T);

ODES  is a list of ordinary differential equations to be converted,
   or a list of labels thereof.
YLIST  is a list of the dependent variables.
T  is the independent variable.

For convenience, square brackets may be omitted from one-element lists.