49Nxx Geometry and constraints
This subtopic introduces the core ideas in geometry and constraints, including foundational concepts, standard methods, and the main questions used to organize the area. Typical uses include building mathematical background, framing related research problems, and supporting applications in neighboring fields where these concepts provide useful structure.
Specific topics
49N05 Linear optimal control problems
Overview
Linear optimal control problems exploit linear dynamics and convex costs to obtain tractable theory and efficient computational methods.
Related Wikipedia Page
Wikipedia: Linear-quadratic regulator
Useful Links
Key Ideas
- linear dynamics
- convex cost
- feedback laws
Typical Uses
Central in engineering control design.
Applications
- Aerospace control
- Process control
- Robotics
References
Recommended Textbooks
49N10 Linear-quadratic optimal control problems
Overview
The linear-quadratic framework gives closed-form solutions for many optimal control problems and acts as a benchmark for more general theory.
Related Wikipedia Page
Wikipedia: Linear-quadratic regulator
Useful Links
Key Ideas
- Riccati equations
- quadratic cost
- feedback gains
Typical Uses
Widely used in aerospace, robotics, and process control.
Applications
- Satellite attitude control
- Vehicle stabilization
- Supply chains
References
Recommended Textbooks
49N15 Duality theory in mathematical programming
Overview
Duality theory relates a primal optimization problem to a related dual problem and gives powerful certificates of optimality and sensitivity.
Related Wikipedia Page
Wikipedia: Duality (optimization)
Useful Links
Key Ideas
- primal-dual relationships
- weak and strong duality
- sensitivity
Typical Uses
Used in economics, operations research, and convex analysis.
Applications
- Network optimization
- Pricing
- Resource allocation
References
Recommended Textbooks
49N20 Periodic optimal control problems
Overview
This topic studies control problems with periodic objectives, constraints, or state trajectories, where periodicity imposes special structure.
Related Wikipedia Page
Wikipedia: Periodic orbit
Useful Links
Key Ideas
- periodic trajectories
- Floquet theory
- cyclic optimization
Typical Uses
Relevant in cyclic processes and repetitive control tasks.
Applications
- Energy management
- Manufacturing cycles
- Biological rhythms
References
Recommended Textbooks
49N25 Impulsive optimal control problems
Overview
Impulsive control problems allow instantaneous changes in the state, making them suitable for models with jumps or abrupt interventions.
Related Wikipedia Page
Impulsive control (Wikipedia)
Useful Links
Key Ideas
- jump dynamics
- impulse controls
- hybrid systems
Typical Uses
Applied when decisions happen in discrete bursts rather than continuously.
Applications
- Finance
- Epidemiology
- Hybrid systems
References
Recommended Textbooks
49N30 Problems with incomplete information, min-max optimization, robust control
Overview
This area studies decision problems where information is incomplete or uncertain, and the objective is to perform well under worst-case scenarios.
Related Wikipedia Page
Wikipedia: Robust control
Useful Links
Key Ideas
- min-max formulations
- robustness
- uncertainty sets
Typical Uses
Standard in systems engineering and decision theory under uncertainty.
Applications
- Robotics
- Power systems
- Finance
References
Recommended Textbooks
49N35 Optimal feedback synthesis
Overview
Optimal feedback synthesis seeks control laws that depend on the current state and implement the optimal policy dynamically.
Related Wikipedia Page
Wikipedia: Feedback
Useful Links
Key Ideas
- feedback laws
- Hamilton-Jacobi-Bellman equations
- stabilization
Typical Uses
Central in control engineering and robotics.
Applications
- Autonomous vehicles
- Process control
- Aerospace guidance
References
Recommended Textbooks
49N45 Inverse problems in optimal control
Overview
Inverse problems in optimal control seek to recover unknown parameters, costs, or constraints from observed behavior or measurements.
Related Wikipedia Page
Wikipedia: Inverse problem
Useful Links
Key Ideas
- parameter identification
- regularization
- data assimilation
Typical Uses
Used in systems biology, geophysics, and engineering diagnostics.
Applications
- Medical imaging
- Geophysics
- Parameter estimation
References
Recommended Textbooks
49N60 Regularity of solutions/controls
Overview
Regularity theory studies smoothness and structural properties of optimal controls and their associated states or value functions.
Related Wikipedia Page
Wikipedia: Regularity theory
Useful Links
Key Ideas
- smoothness
- boundary regularity
- Sobolev regularity
Typical Uses
Important for numerical approximation and qualitative analysis.
Applications
- PDE control
- Free-boundary problems
- Mechanics
References
Recommended Textbooks
49N70 Differential games and control
Overview
Differential games model interactions among multiple agents, each pursuing their own objective under dynamic constraints.
Related Wikipedia Page
Wikipedia: Differential game
Useful Links
Key Ideas
- Nash equilibria
- feedback strategies
- value functions
Typical Uses
Used in economics, robotics, and pursuit-evasion problems.
Applications
- Pursuit-evasion
- Defense problems
- Market competition
References
Recommended Textbooks
49N75 Pursuit and evasion games
Overview
Pursuit-evasion problems study how one or more pursuers can catch an evader while both respond dynamically.
Related Wikipedia Page
Wikipedia: Pursuit-evasion
Useful Links
Key Ideas
- capture conditions
- state constraints
- strategic interaction
Typical Uses
Relevant to surveillance, robotics, and security.
Applications
- Search-and-rescue
- Robotics
- Defense
References
Recommended Textbooks
49N80 Mean field games and control
Overview
Mean field methods approximate large-population interactions by a continuum of agents and study equilibria and optimization in that limit.
Related Wikipedia Page
Wikipedia: Mean field game theory
Useful Links
Key Ideas
- mean field equilibria
- coupled PDE systems
- population limits
Typical Uses
Used in economics, traffic modeling, and swarm control.
Applications
- Traffic flow
- Crowd dynamics
- Finance
References
Recommended Textbooks
49N90 Applications of optimal control and differential games
Overview
This topic studies how the theory of optimal control and differential games is applied to real-world systems across engineering, economics, and biology.
Related Wikipedia Page
Wikipedia: Optimal control
Useful Links
Key Ideas
- application-driven modeling
- interdisciplinary methods
- implementation challenges
Typical Uses
Useful as a bridge between theory and practice in many applied domains.
Applications
- Biological control
- Energy systems
- Transportation
References
Recommended Textbooks