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Problem Types

This module defines a set of structures used to represent different control problems.

All problems are subtypes of the abstract type ProblemType, which provides a common interface for control problems.

Supported Problems

  • EmptyProblem: Used to construct an abstraction of a dynamical system without solving a control problem.

  • OptimalControlProblem: A reach-avoid optimal control problem defined over a finite time horizon, supporting state and transition costs.

  • SafetyProblem: A safety specification problem requiring the system to remain within a safe set for the entire time horizon.

Each of these problem types is detailed below:

Dionysos.Problem.EmptyProblem — Type
EmptyProblem{S, X} <: ProblemType

A problem type used to construct an abstraction of a dynamical system.

  • S: The system to abstract (continuous or discrete-time).
  • X: The region of interest (e.g., a subset of the state space).

This problem encodes no control objective. It is intended for generating symbolic models that can later be reused by other solvers.

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Dionysos.Problem.OptimalControlProblem — Type
OptimalControlProblem{S, XI, XT, XC, TC, T} <: ProblemType

Encodes a reach-avoid optimal control problem over a finite horizon.

  • S: The system to control.
  • XI: The initial set of states.
  • XT: The target set to be reached.
  • XC: A state cost function or structure.
  • TC: A transition cost function or structure.
  • T: The time horizon (number of allowed time steps).

This problem aims to find a control strategy that reaches the target set from the initial set, minimizing the accumulated cost over time.

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Dionysos.Problem.SafetyProblem — Type
SafetyProblem{S, XI, XS, T} <: ProblemType

Encodes a safety control problem over a finite horizon.

  • S: The system to control.
  • XI: The initial set of states.
  • XS: The safe set in which the system must remain.
  • T: The time horizon (number of allowed time steps).

This problem aims to synthesize a controller that ensures the system remains within the safe set for the entire duration of the time horizon.

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