Slutvillkor

Slutvillkor is a term used in Swedish mathematics and applied sciences to describe constraints imposed on the solution of a problem at the boundaries of the domain or at a specified final time. In differential equations and related areas, these constraints are typically referred to as boundary conditions or terminal conditions, depending on whether they apply to spatial boundaries or to a time endpoint.

In spatial problems, slutvilkor specify the values or behavior of the unknown function on the boundary of the domain. Common types include Dirichlet conditions, which prescribe the function’s value on the boundary; Neumann conditions, which prescribe the normal derivative; and Robin conditions, which combine the function and its derivative linearly. These conditions are essential to determine a unique solution to a boundary value problem.

In time-dependent problems, slutvilkor define the state of the system at a final time. This appears in terminal or final conditions in, for example, optimal control and calculus of variations, where the endpoint constraint may fix the final state or include a terminal cost function that influences the optimal trajectory.

Examples include the heat or wave equations on a region with specified boundary values (Dirichlet), specified flux across the boundary (Neumann), or a mixture of these (Robin). In optimal control, one might require the state to reach a target at time T or minimize a cost at T.

Existence and uniqueness results, numerical discretization, and stability analyses all rely on the appropriate specification of slutvillkor, reflecting the physical or practical constraints of the problem. See also boundary value problems, terminal value problems, and initial value problems.