Large Transport
Air Traffic Control Fully Certified
Air Traffic Control Time Certified In Pos 1 (yrs) 5
Confusion; Situational Awareness
Airspace Violation All Types; Deviation / Discrepancy - Procedural Clearance
Air Traffic Control Provided Assistance; Flight Crew Returned To Clearance
Human Factors; Procedure; Weather; Airspace Structure
Aircraft X departed RWY01 and was radar identified and climbed to 17000.  He continued runway heading off the SID and violated P56B.  As he continued to climb it appeared that he was going to violate P56A.  I turned the aircraft right to 040 to avoid but the pilot questioned it.  Aircraft X entered P56A.Give PCT the southwest part of P56 for departures.
Standard preflight at the gate.  Addressed a couple of issues with ground personnel but nothing distracting (aircraft was powered upon arrival and another small issue that escapes me).  Taxi out was uneventful and not rushed.  It was still mostly dark on departure.  We departed on the NATIONAL 3 SID.  I was the Pilot Monitoring.  At 400 feet; I asked the Pilot flying if he wanted LNAV and then selected LNAV.  He said no; VOR/LOC.  So; I deselected LNAV and selected VOR/LOC.  We had entered IMC conditions around 400 feet AGL. The pilot Flying started making a gradual left turn toward our outbound course of 328 degrees.  Although it was a slower turn than I preferred; I wrongly assumed the Pilot flying was a bit disoriented and was making a gradual turn to avoid getting more disoriented.  The turn was consistent although shallow so I did not say anything.  I also thought that if he is disoriented; the last thing I should do is start directing him and breaking his concentration.  He continued to hand-fly the departure. Our clearance was to 5000 feet MSL.  Passing through 4000 feet; I felt comfortable enough to tell him we needed to get back on course.  At that time; Potomac Departure called and said for an immediate left turn.  Since he was still hand-flying in VOR/LOC capture; I put my finger up on the Mode Control Panel; stated Heading Select; he concurred; and I selected Heading Select.  Almost immediately after that; Potomac Departure changed our heading to a right turn of 040. We leveled at 5000 feet and followed ATC instructions. Above 10;000; we discussed what each saw.  I don't want to misquote the Pilot flying but I thought he said he was not disoriented; just concentrating flying the flight director.  I on the other hand; was looking at the RNAV magenta course of 333-degrees to BUFFR.It was a very long and quiet ride to [destination].  We were met at the [destination] gate by OSI; TSA; and FAA representatives. They were cordial; said they needed to ask us a few questions; and assured us we were not being arrested today.  At this point; I contacted the Chief Pilot on Call and said we were not in a mental position to continue flying.  Both Company and Union representatives were extremely helpful.  It is an ongoing investigation so I don't know how much to share. Lessons learned?  Since the airspace at DCA is too tight; we should have discussed ahead of time which roll mode would be used.  The pilot flying was legal to call for VOR/LOC so we could track out on the 328 radial of DCA.  Second; I may have induced a flight director error by selecting LNAV and then deselected and immediately selecting VOR/LOC.  I do not know if there is a lag time for the flight director to catch-up. Even my idea of flying LNAV off the 333 radial to BUFFR doesn't match the SID clearance of 328 course off DCA.
PCT Controller and pilots of the aircraft describe an airspace violation.