Nonsingular Problems Research Articles

Error bounds for the Galerkin method applied to singular and nonsingular boundary value problems

The purpose of this article is to obtainL 2 and uniform norm error estimates for the Galerkin approximation of the solution of certain boundary value problems via a comparison with then-norm projection of the solution. In some cases, these estimates constitute an improvement over known results.

Sufficient Conditions for Nonnegativity of the Second Variation in Singular and Nonsingular Control Problems

Sufficient conditions for nonnegativity of the second variation in singular and nonsingular control problems are presented; these conditions are in the form of equalities and differential inequalities. Control problem examples illustrate the use of the new conditions. The relationships of the new conditions to existing necessary conditions of optimality for singular and nonsingular problems are discussed. When applied to nonsingular control problems, it is shown that the conditions are sufficient to ensure the boundedness of the solution of the well-known matrix Riccati differential equation.

Computation of optimal singular controls

A class of singular control problems is made nonsingular by the addition of an integral quadratic functional of the control to the cost functional; a parameter <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon &gt; 0</tex> multiplies this added functional. The resulting nonsingular problem is solved for a monotonically decreasing sequence <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{\epsilon; \epsilon_{1} &gt; \epsilon_{2} &gt; ... &gt; \epsilon_{k} &gt; 0\}</tex> . As <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k \rightarrow \infty</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon_{k} \rightarrow 0</tex> the solution of the modified problem tends to the solution of the original singular problem. A variant of the method which does not require that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon \rightarrow 0</tex> is also presented. Four illustrative numerical examples are described.

Discrete states for non-singular and singular potential problems

Discrete states for non-singular and singular potential problems

Definitely self-adjoint boundary value problems

The system which is given in (1.1) is said to be self-adjoint provided that it is equivalent to its adjoint system (1.2) by a non-singular transformation zi = Tia(X)ya. This definition of self-adjoint boundary value problems and a further definition of so-called definite self-adjointness were given by the author in a paper published in 1926t which will be designated in the text below by the Roman numeral I. in that paper it was stated that the boundary value problems arising from the calculus of variations are all definitely self-adjoint. This statement is true for non-singular problems of the calculus of variations without side conditions, the only ones whose boundary value problems had been studied up to that time so far as is known to the writer. It is not true, however, for problems of the calculus of variations such as those of Mayer, Lagrange, and Bolza whose boundary value problems are self-adjoint but not definitely self-adjoint according to the definition given in I. One of the earliest